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Theorem fsuppind 39443
Description: Induction on functions 𝐹:𝐴𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 20448 and frlmplusgval 20456). This theorem is structurally general for polynomial proof usage (see mplelbas 20671 and mpladd 20683). (Contributed by SN, 18-May-2024.)
Hypotheses
Ref Expression
fsuppind.b 𝐵 = (Base‘𝐺)
fsuppind.z 0 = (0g𝐺)
fsuppind.p + = (+g𝐺)
fsuppind.g (𝜑𝐺 ∈ Grp)
fsuppind.v (𝜑𝐼𝑉)
fsuppind.0 (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
fsuppind.1 ((𝜑 ∧ (𝑎𝐼𝑏𝐵)) → (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
fsuppind.2 ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)
Assertion
Ref Expression
fsuppind ((𝜑 ∧ (𝑋:𝐼𝐵𝑋 finSupp 0 )) → 𝑋𝐻)
Distinct variable groups:   𝑥, + ,𝑦   0 ,𝑎,𝑏,𝑥   𝑦, 0   𝐼,𝑎,𝑏,𝑥   𝑦,𝐼   𝐻,𝑏   𝑦,𝐻,𝑥   𝐻,𝑎   𝜑,𝑥,𝑦   𝜑,𝑎,𝑏   𝐵,𝑎,𝑏,𝑥
Allowed substitution hints:   𝐵(𝑦)   + (𝑎,𝑏)   𝐺(𝑥,𝑦,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑎,𝑏)   𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem fsuppind
Dummy variables 𝑧 𝑐 𝑚 𝑣 𝑖 𝑗 𝑛 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsuppind.b . . . . . . . . . . 11 𝐵 = (Base‘𝐺)
21fvexi 6663 . . . . . . . . . 10 𝐵 ∈ V
32a1i 11 . . . . . . . . 9 (𝜑𝐵 ∈ V)
4 fsuppind.v . . . . . . . . 9 (𝜑𝐼𝑉)
53, 4elmapd 8407 . . . . . . . 8 (𝜑 → (𝑋 ∈ (𝐵m 𝐼) ↔ 𝑋:𝐼𝐵))
65adantr 484 . . . . . . 7 ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋 ∈ (𝐵m 𝐼) ↔ 𝑋:𝐼𝐵))
7 eqeq1 2805 . . . . . . . . . . . . . . . 16 (𝑖 = 1 → (𝑖 = (♯‘( supp 0 )) ↔ 1 = (♯‘( supp 0 ))))
87imbi1d 345 . . . . . . . . . . . . . . 15 (𝑖 = 1 → ((𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ (1 = (♯‘( supp 0 )) → 𝐻)))
98ralbidv 3165 . . . . . . . . . . . . . 14 (𝑖 = 1 → (∀ ∈ (𝐵m 𝐼)(𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ ∀ ∈ (𝐵m 𝐼)(1 = (♯‘( supp 0 )) → 𝐻)))
10 eqeq1 2805 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑗 → (𝑖 = (♯‘( supp 0 )) ↔ 𝑗 = (♯‘( supp 0 ))))
1110imbi1d 345 . . . . . . . . . . . . . . 15 (𝑖 = 𝑗 → ((𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ (𝑗 = (♯‘( supp 0 )) → 𝐻)))
1211ralbidv 3165 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (∀ ∈ (𝐵m 𝐼)(𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)))
13 eqeq1 2805 . . . . . . . . . . . . . . . 16 (𝑖 = (𝑗 + 1) → (𝑖 = (♯‘( supp 0 )) ↔ (𝑗 + 1) = (♯‘( supp 0 ))))
1413imbi1d 345 . . . . . . . . . . . . . . 15 (𝑖 = (𝑗 + 1) → ((𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ ((𝑗 + 1) = (♯‘( supp 0 )) → 𝐻)))
1514ralbidv 3165 . . . . . . . . . . . . . 14 (𝑖 = (𝑗 + 1) → (∀ ∈ (𝐵m 𝐼)(𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ ∀ ∈ (𝐵m 𝐼)((𝑗 + 1) = (♯‘( supp 0 )) → 𝐻)))
16 eqeq1 2805 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑛 → (𝑖 = (♯‘( supp 0 )) ↔ 𝑛 = (♯‘( supp 0 ))))
1716imbi1d 345 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → ((𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ (𝑛 = (♯‘( supp 0 )) → 𝐻)))
1817ralbidv 3165 . . . . . . . . . . . . . 14 (𝑖 = 𝑛 → (∀ ∈ (𝐵m 𝐼)(𝑖 = (♯‘( supp 0 )) → 𝐻) ↔ ∀ ∈ (𝐵m 𝐼)(𝑛 = (♯‘( supp 0 )) → 𝐻)))
19 eqcom 2808 . . . . . . . . . . . . . . . . 17 (1 = (♯‘( supp 0 )) ↔ (♯‘( supp 0 )) = 1)
20 ovex 7172 . . . . . . . . . . . . . . . . . 18 ( supp 0 ) ∈ V
21 euhash1 13781 . . . . . . . . . . . . . . . . . 18 (( supp 0 ) ∈ V → ((♯‘( supp 0 )) = 1 ↔ ∃!𝑐 𝑐 ∈ ( supp 0 )))
2220, 21ax-mp 5 . . . . . . . . . . . . . . . . 17 ((♯‘( supp 0 )) = 1 ↔ ∃!𝑐 𝑐 ∈ ( supp 0 ))
2319, 22bitri 278 . . . . . . . . . . . . . . . 16 (1 = (♯‘( supp 0 )) ↔ ∃!𝑐 𝑐 ∈ ( supp 0 ))
24 elmapfn 8416 . . . . . . . . . . . . . . . . . . . . 21 ( ∈ (𝐵m 𝐼) → Fn 𝐼)
2524adantl 485 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∈ (𝐵m 𝐼)) → Fn 𝐼)
264adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∈ (𝐵m 𝐼)) → 𝐼𝑉)
27 fsuppind.z . . . . . . . . . . . . . . . . . . . . . 22 0 = (0g𝐺)
2827fvexi 6663 . . . . . . . . . . . . . . . . . . . . 21 0 ∈ V
2928a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∈ (𝐵m 𝐼)) → 0 ∈ V)
30 elsuppfn 7825 . . . . . . . . . . . . . . . . . . . 20 (( Fn 𝐼𝐼𝑉0 ∈ V) → (𝑐 ∈ ( supp 0 ) ↔ (𝑐𝐼 ∧ (𝑐) ≠ 0 )))
3125, 26, 29, 30syl3anc 1368 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∈ (𝐵m 𝐼)) → (𝑐 ∈ ( supp 0 ) ↔ (𝑐𝐼 ∧ (𝑐) ≠ 0 )))
3231eubidv 2650 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∈ (𝐵m 𝐼)) → (∃!𝑐 𝑐 ∈ ( supp 0 ) ↔ ∃!𝑐(𝑐𝐼 ∧ (𝑐) ≠ 0 )))
33 df-reu 3116 . . . . . . . . . . . . . . . . . 18 (∃!𝑐𝐼 (𝑐) ≠ 0 ↔ ∃!𝑐(𝑐𝐼 ∧ (𝑐) ≠ 0 ))
3432, 33syl6bbr 292 . . . . . . . . . . . . . . . . 17 ((𝜑 ∈ (𝐵m 𝐼)) → (∃!𝑐 𝑐 ∈ ( supp 0 ) ↔ ∃!𝑐𝐼 (𝑐) ≠ 0 ))
3524ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → Fn 𝐼)
36 fvex 6662 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥) ∈ V
3736, 28ifex 4476 . . . . . . . . . . . . . . . . . . . . . 22 if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 ) ∈ V
38 eqid 2801 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )) = (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 ))
3937, 38fnmpti 6467 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )) Fn 𝐼
4039a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )) Fn 𝐼)
41 eqeq1 2805 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑣 → (𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ) ↔ 𝑣 = (𝑐𝐼 (𝑐) ≠ 0 )))
42 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑣 → (𝑥) = (𝑣))
4341, 42ifbieq1d 4451 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑣 → if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 ) = if(𝑣 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑣), 0 ))
4443, 38, 37fvmpt3i 6754 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝐼 → ((𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 ))‘𝑣) = if(𝑣 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑣), 0 ))
4544adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → ((𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 ))‘𝑣) = if(𝑣 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑣), 0 ))
46 eqidd 2802 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) ∧ 𝑣 = (𝑐𝐼 (𝑐) ≠ 0 )) → (𝑣) = (𝑣))
47 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → 𝑣𝐼)
48 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → ∃!𝑐𝐼 (𝑐) ≠ 0 )
49 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑐 = 𝑣 → (𝑐) = (𝑣))
5049neeq1d 3049 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = 𝑣 → ((𝑐) ≠ 0 ↔ (𝑣) ≠ 0 ))
5150riota2 7122 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑣𝐼 ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → ((𝑣) ≠ 0 ↔ (𝑐𝐼 (𝑐) ≠ 0 ) = 𝑣))
5247, 48, 51syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → ((𝑣) ≠ 0 ↔ (𝑐𝐼 (𝑐) ≠ 0 ) = 𝑣))
53 necom 3043 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ( 0 ≠ (𝑣) ↔ (𝑣) ≠ 0 )
54 eqcom 2808 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑣 = (𝑐𝐼 (𝑐) ≠ 0 ) ↔ (𝑐𝐼 (𝑐) ≠ 0 ) = 𝑣)
5552, 53, 543bitr4g 317 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → ( 0 ≠ (𝑣) ↔ 𝑣 = (𝑐𝐼 (𝑐) ≠ 0 )))
5655biimpd 232 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → ( 0 ≠ (𝑣) → 𝑣 = (𝑐𝐼 (𝑐) ≠ 0 )))
5756necon1bd 3008 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → (¬ 𝑣 = (𝑐𝐼 (𝑐) ≠ 0 ) → 0 = (𝑣)))
5857imp 410 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) ∧ ¬ 𝑣 = (𝑐𝐼 (𝑐) ≠ 0 )) → 0 = (𝑣))
5946, 58ifeqda 4463 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → if(𝑣 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑣), 0 ) = (𝑣))
6045, 59eqtr2d 2837 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) ∧ 𝑣𝐼) → (𝑣) = ((𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 ))‘𝑣))
6135, 40, 60eqfnfvd 6786 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → = (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )))
62 riotacl 7114 . . . . . . . . . . . . . . . . . . . . 21 (∃!𝑐𝐼 (𝑐) ≠ 0 → (𝑐𝐼 (𝑐) ≠ 0 ) ∈ 𝐼)
6362adantl 485 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → (𝑐𝐼 (𝑐) ≠ 0 ) ∈ 𝐼)
64 elmapi 8415 . . . . . . . . . . . . . . . . . . . . . 22 ( ∈ (𝐵m 𝐼) → :𝐼𝐵)
6564ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → :𝐼𝐵)
6665, 63ffvelrnd 6833 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → (‘(𝑐𝐼 (𝑐) ≠ 0 )) ∈ 𝐵)
67 fsuppind.1 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑎𝐼𝑏𝐵)) → (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
6867ralrimivva 3159 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑎𝐼𝑏𝐵 (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
6968ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → ∀𝑎𝐼𝑏𝐵 (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
70 eqeq2 2813 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = (𝑐𝐼 (𝑐) ≠ 0 ) → (𝑥 = 𝑎𝑥 = (𝑐𝐼 (𝑐) ≠ 0 )))
7170ifbid 4450 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = (𝑐𝐼 (𝑐) ≠ 0 ) → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), 𝑏, 0 ))
7271mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = (𝑐𝐼 (𝑐) ≠ 0 ) → (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), 𝑏, 0 )))
7372eleq1d 2877 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑐𝐼 (𝑐) ≠ 0 ) → ((𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), 𝑏, 0 )) ∈ 𝐻))
74 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ) → (𝑥) = (‘(𝑐𝐼 (𝑐) ≠ 0 )))
7574eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ) → (𝑏 = (𝑥) ↔ 𝑏 = (‘(𝑐𝐼 (𝑐) ≠ 0 ))))
7675biimparc 483 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑏 = (‘(𝑐𝐼 (𝑐) ≠ 0 )) ∧ 𝑥 = (𝑐𝐼 (𝑐) ≠ 0 )) → 𝑏 = (𝑥))
7776ifeq1da 4458 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (‘(𝑐𝐼 (𝑐) ≠ 0 )) → if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), 𝑏, 0 ) = if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 ))
7877mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (‘(𝑐𝐼 (𝑐) ≠ 0 )) → (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), 𝑏, 0 )) = (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )))
7978eleq1d 2877 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (‘(𝑐𝐼 (𝑐) ≠ 0 )) → ((𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )) ∈ 𝐻))
8073, 79rspc2va 3585 . . . . . . . . . . . . . . . . . . . 20 ((((𝑐𝐼 (𝑐) ≠ 0 ) ∈ 𝐼 ∧ (‘(𝑐𝐼 (𝑐) ≠ 0 )) ∈ 𝐵) ∧ ∀𝑎𝐼𝑏𝐵 (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻) → (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )) ∈ 𝐻)
8163, 66, 69, 80syl21anc 836 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → (𝑥𝐼 ↦ if(𝑥 = (𝑐𝐼 (𝑐) ≠ 0 ), (𝑥), 0 )) ∈ 𝐻)
8261, 81eqeltrd 2893 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∈ (𝐵m 𝐼)) ∧ ∃!𝑐𝐼 (𝑐) ≠ 0 ) → 𝐻)
8382ex 416 . . . . . . . . . . . . . . . . 17 ((𝜑 ∈ (𝐵m 𝐼)) → (∃!𝑐𝐼 (𝑐) ≠ 0𝐻))
8434, 83sylbid 243 . . . . . . . . . . . . . . . 16 ((𝜑 ∈ (𝐵m 𝐼)) → (∃!𝑐 𝑐 ∈ ( supp 0 ) → 𝐻))
8523, 84syl5bi 245 . . . . . . . . . . . . . . 15 ((𝜑 ∈ (𝐵m 𝐼)) → (1 = (♯‘( supp 0 )) → 𝐻))
8685ralrimiva 3152 . . . . . . . . . . . . . 14 (𝜑 → ∀ ∈ (𝐵m 𝐼)(1 = (♯‘( supp 0 )) → 𝐻))
87 fsuppind.g . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝐺 ∈ Grp)
881, 27grpidcl 18126 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ Grp → 0𝐵)
8987, 88syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑0𝐵)
9089ad5antr 733 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑥𝐼) → 0𝐵)
91 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐵m 𝐼) = (𝐵m 𝐼)
92 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 ∈ (𝐵m 𝐼))
9392ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑥𝐼) → 𝑙 ∈ (𝐵m 𝐼))
94 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑥𝐼) → 𝑥𝐼)
9591, 93, 94mapfvd 8430 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑥𝐼) → (𝑙𝑥) ∈ 𝐵)
9690, 95ifcld 4473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑥𝐼) → if(𝑥 = 𝑧, 0 , (𝑙𝑥)) ∈ 𝐵)
9796fmpttd 6860 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))):𝐼𝐵)
982a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → 𝐵 ∈ V)
994ad4antr 731 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → 𝐼𝑉)
10098, 99elmapd 8407 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∈ (𝐵m 𝐼) ↔ (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))):𝐼𝐵))
10197, 100mpbird 260 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∈ (𝐵m 𝐼))
102101adantrl 715 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∈ (𝐵m 𝐼))
103 fvoveq1 7162 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) → (♯‘(𝑚 supp 0 )) = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )))
104103eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) → (𝑗 = (♯‘(𝑚 supp 0 )) ↔ 𝑗 = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 ))))
105 oveq1 7146 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑚 = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) → (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))) = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))
106105eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑚 = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) → (𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))) ↔ 𝑙 = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))))
107104, 106anbi12d 633 . . . . . . . . . . . . . . . . . . . . . 22 (𝑚 = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))) ↔ (𝑗 = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))))
108107adantl 485 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) ∧ 𝑚 = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))) ↔ (𝑗 = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))))
109 ovexd 7174 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (𝑙 supp 0 ) ∈ V)
110 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝑧𝐼)
111 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (𝑙𝑧) ≠ 0 )
112 elmapfn 8416 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑙 ∈ (𝐵m 𝐼) → 𝑙 Fn 𝐼)
113112ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 Fn 𝐼)
114113adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝑙 Fn 𝐼)
1154ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝐼𝑉)
11628a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 0 ∈ V)
117 elsuppfn 7825 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑙 Fn 𝐼𝐼𝑉0 ∈ V) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )))
118114, 115, 116, 117syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )))
119110, 111, 118mpbir2and 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝑧 ∈ (𝑙 supp 0 ))
120 simpllr 775 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝑗 ∈ ℕ)
121120nnnn0d 11947 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝑗 ∈ ℕ0)
122 simplrr 777 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (𝑗 + 1) = (♯‘(𝑙 supp 0 )))
123122eqcomd 2807 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (♯‘(𝑙 supp 0 )) = (𝑗 + 1))
124 hashdifsnp1 13854 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ0) → ((♯‘(𝑙 supp 0 )) = (𝑗 + 1) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗))
125124imp 410 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ0) ∧ (♯‘(𝑙 supp 0 )) = (𝑗 + 1)) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
126109, 119, 121, 123, 125syl31anc 1370 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
127 eldifsn 4683 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣𝑧))
128 fvex 6662 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑙𝑥) ∈ V
12928, 128ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 if(𝑥 = 𝑧, 0 , (𝑙𝑥)) ∈ V
130 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))
131129, 130fnmpti 6467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) Fn 𝐼
132131a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) Fn 𝐼)
1334ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → 𝐼𝑉)
13428a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → 0 ∈ V)
135 elsuppfn 7825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) Fn 𝐼𝐼𝑉0 ∈ V) → (𝑣 ∈ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 ) ↔ (𝑣𝐼 ∧ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))‘𝑣) ≠ 0 )))
136132, 133, 134, 135syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (𝑣 ∈ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 ) ↔ (𝑣𝐼 ∧ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))‘𝑣) ≠ 0 )))
137 iftrue 4434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙𝑣)) = 0 )
138 olc 865 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑣 = 𝑧 → ((𝑙𝑣) = 0𝑣 = 𝑧))
139137, 1382thd 268 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) = 0 ↔ ((𝑙𝑣) = 0𝑣 = 𝑧)))
140 iffalse 4437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙𝑣)) = (𝑙𝑣))
141140eqeq1d 2803 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) = 0 ↔ (𝑙𝑣) = 0 ))
142 biorf 934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑣 = 𝑧 → ((𝑙𝑣) = 0 ↔ (𝑣 = 𝑧 ∨ (𝑙𝑣) = 0 )))
143 orcom 867 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑙𝑣) = 0𝑣 = 𝑧) ↔ (𝑣 = 𝑧 ∨ (𝑙𝑣) = 0 ))
144142, 143syl6bbr 292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 𝑣 = 𝑧 → ((𝑙𝑣) = 0 ↔ ((𝑙𝑣) = 0𝑣 = 𝑧)))
145141, 144bitrd 282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) = 0 ↔ ((𝑙𝑣) = 0𝑣 = 𝑧)))
146139, 145pm2.61i 185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) = 0 ↔ ((𝑙𝑣) = 0𝑣 = 𝑧))
147146a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) = 0 ↔ ((𝑙𝑣) = 0𝑣 = 𝑧)))
148147necon3abid 3026 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) ≠ 0 ↔ ¬ ((𝑙𝑣) = 0𝑣 = 𝑧)))
149 neanior 3082 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((𝑙𝑣) ≠ 0𝑣𝑧) ↔ ¬ ((𝑙𝑣) = 0𝑣 = 𝑧))
150148, 149syl6bbr 292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) ≠ 0 ↔ ((𝑙𝑣) ≠ 0𝑣𝑧)))
151150anbi2d 631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑣𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙𝑣)) ≠ 0 ) ↔ (𝑣𝐼 ∧ ((𝑙𝑣) ≠ 0𝑣𝑧))))
152 anass 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑣𝐼 ∧ (𝑙𝑣) ≠ 0 ) ∧ 𝑣𝑧) ↔ (𝑣𝐼 ∧ ((𝑙𝑣) ≠ 0𝑣𝑧)))
153151, 152syl6bbr 292 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑣𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙𝑣)) ≠ 0 ) ↔ ((𝑣𝐼 ∧ (𝑙𝑣) ≠ 0 ) ∧ 𝑣𝑧)))
154 equequ1 2032 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 = 𝑣 → (𝑥 = 𝑧𝑣 = 𝑧))
155 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑥 = 𝑣 → (𝑙𝑥) = (𝑙𝑣))
156154, 155ifbieq2d 4453 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑥 = 𝑣 → if(𝑥 = 𝑧, 0 , (𝑙𝑥)) = if(𝑣 = 𝑧, 0 , (𝑙𝑣)))
157156, 130, 129fvmpt3i 6754 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑣𝐼 → ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))‘𝑣) = if(𝑣 = 𝑧, 0 , (𝑙𝑣)))
158157adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))‘𝑣) = if(𝑣 = 𝑧, 0 , (𝑙𝑣)))
159158neeq1d 3049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → (((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))‘𝑣) ≠ 0 ↔ if(𝑣 = 𝑧, 0 , (𝑙𝑣)) ≠ 0 ))
160159pm5.32da 582 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑣𝐼 ∧ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))‘𝑣) ≠ 0 ) ↔ (𝑣𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙𝑣)) ≠ 0 )))
161113adantr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → 𝑙 Fn 𝐼)
162 elsuppfn 7825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑙 Fn 𝐼𝐼𝑉0 ∈ V) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣𝐼 ∧ (𝑙𝑣) ≠ 0 )))
163161, 133, 134, 162syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣𝐼 ∧ (𝑙𝑣) ≠ 0 )))
164163anbi1d 632 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣𝑧) ↔ ((𝑣𝐼 ∧ (𝑙𝑣) ≠ 0 ) ∧ 𝑣𝑧)))
165153, 160, 1643bitr4d 314 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑣𝐼 ∧ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥)))‘𝑣) ≠ 0 ) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣𝑧)))
166136, 165bitr2d 283 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣𝑧) ↔ 𝑣 ∈ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )))
167127, 166syl5bb 286 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ 𝑣 ∈ ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )))
168167eqrdv 2799 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑙 supp 0 ) ∖ {𝑧}) = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 ))
169168fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )))
170169adantrl 715 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )))
171126, 170eqtr3d 2838 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝑗 = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )))
172128, 28ifex 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 if(𝑥 = 𝑧, (𝑙𝑥), 0 ) ∈ V
173 eqid 2801 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )) = (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))
174172, 173fnmpti 6467 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )) Fn 𝐼
175174a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )) Fn 𝐼)
176 inidm 4148 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝐼𝐼) = 𝐼
177132, 175, 133, 133, 176offn 7404 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))) Fn 𝐼)
178154, 155ifbieq1d 4451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑣 → if(𝑥 = 𝑧, (𝑙𝑥), 0 ) = if(𝑣 = 𝑧, (𝑙𝑣), 0 ))
179178, 173, 172fvmpt3i 6754 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑣𝐼 → ((𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))‘𝑣) = if(𝑣 = 𝑧, (𝑙𝑣), 0 ))
180179adantl 485 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → ((𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))‘𝑣) = if(𝑣 = 𝑧, (𝑙𝑣), 0 ))
181132, 175, 133, 133, 176, 158, 180ofval 7402 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → (((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))‘𝑣) = (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) + if(𝑣 = 𝑧, (𝑙𝑣), 0 )))
18287ad4antr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → 𝐺 ∈ Grp)
183 simplrl 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ ((𝑙𝑧) ≠ 0𝑣𝐼)) → 𝑙 ∈ (𝐵m 𝐼))
184183anassrs 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → 𝑙 ∈ (𝐵m 𝐼))
185 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → 𝑣𝐼)
18691, 184, 185mapfvd 8430 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → (𝑙𝑣) ∈ 𝐵)
187 fsuppind.p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 + = (+g𝐺)
1881, 187, 27grplid 18128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺 ∈ Grp ∧ (𝑙𝑣) ∈ 𝐵) → ( 0 + (𝑙𝑣)) = (𝑙𝑣))
1891, 187, 27grprid 18129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐺 ∈ Grp ∧ (𝑙𝑣) ∈ 𝐵) → ((𝑙𝑣) + 0 ) = (𝑙𝑣))
190188, 189ifeq12d 4448 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝐺 ∈ Grp ∧ (𝑙𝑣) ∈ 𝐵) → if(𝑣 = 𝑧, ( 0 + (𝑙𝑣)), ((𝑙𝑣) + 0 )) = if(𝑣 = 𝑧, (𝑙𝑣), (𝑙𝑣)))
191182, 186, 190syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → if(𝑣 = 𝑧, ( 0 + (𝑙𝑣)), ((𝑙𝑣) + 0 )) = if(𝑣 = 𝑧, (𝑙𝑣), (𝑙𝑣)))
192 ovif12 7236 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) + if(𝑣 = 𝑧, (𝑙𝑣), 0 )) = if(𝑣 = 𝑧, ( 0 + (𝑙𝑣)), ((𝑙𝑣) + 0 ))
193 ifid 4467 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 if(𝑣 = 𝑧, (𝑙𝑣), (𝑙𝑣)) = (𝑙𝑣)
194193eqcomi 2810 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑙𝑣) = if(𝑣 = 𝑧, (𝑙𝑣), (𝑙𝑣))
195191, 192, 1943eqtr4g 2861 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → (if(𝑣 = 𝑧, 0 , (𝑙𝑣)) + if(𝑣 = 𝑧, (𝑙𝑣), 0 )) = (𝑙𝑣))
196181, 195eqtr2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) ∧ 𝑣𝐼) → (𝑙𝑣) = (((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))‘𝑣))
197161, 177, 196eqfnfvd 6786 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙𝑧) ≠ 0 ) → 𝑙 = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))
198197adantrl 715 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → 𝑙 = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))
199171, 198jca 515 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (𝑗 = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))))
200199adantllr 718 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → (𝑗 = (♯‘((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙𝑥))) ∘f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))))
201102, 108, 200rspcedvd 3577 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧𝐼 ∧ (𝑙𝑧) ≠ 0 )) → ∃𝑚 ∈ (𝐵m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))))
202112ad2antrl 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 Fn 𝐼)
2034ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝐼𝑉)
20428a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 0 ∈ V)
205 suppvalfn 7824 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 Fn 𝐼𝐼𝑉0 ∈ V) → (𝑙 supp 0 ) = {𝑧𝐼 ∣ (𝑙𝑧) ≠ 0 })
206202, 203, 204, 205syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑙 supp 0 ) = {𝑧𝐼 ∣ (𝑙𝑧) ≠ 0 })
207 simprr 772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) = (♯‘(𝑙 supp 0 )))
208 peano2nn 11641 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
209208ad3antlr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) ∈ ℕ)
210209nnne0d 11679 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) ≠ 0)
211207, 210eqnetrrd 3058 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (♯‘(𝑙 supp 0 )) ≠ 0)
212 ovex 7172 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 supp 0 ) ∈ V
213 hasheq0 13724 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) = 0 ↔ (𝑙 supp 0 ) = ∅))
214213necon3bid 3034 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) ≠ 0 ↔ (𝑙 supp 0 ) ≠ ∅))
215212, 214mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ((♯‘(𝑙 supp 0 )) ≠ 0 ↔ (𝑙 supp 0 ) ≠ ∅))
216211, 215mpbid 235 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑙 supp 0 ) ≠ ∅)
217206, 216eqnetrrd 3058 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → {𝑧𝐼 ∣ (𝑙𝑧) ≠ 0 } ≠ ∅)
218 rabn0 4296 . . . . . . . . . . . . . . . . . . . . 21 ({𝑧𝐼 ∣ (𝑙𝑧) ≠ 0 } ≠ ∅ ↔ ∃𝑧𝐼 (𝑙𝑧) ≠ 0 )
219217, 218sylib 221 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑧𝐼 (𝑙𝑧) ≠ 0 )
220201, 219reximddv 3237 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑧𝐼𝑚 ∈ (𝐵m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))))
221 rexcom 3311 . . . . . . . . . . . . . . . . . . 19 (∃𝑧𝐼𝑚 ∈ (𝐵m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))) ↔ ∃𝑚 ∈ (𝐵m 𝐼)∃𝑧𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))))
222220, 221sylib 221 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑚 ∈ (𝐵m 𝐼)∃𝑧𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))))
223 simprr 772 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))
224 fvoveq1 7162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ( = 𝑚 → (♯‘( supp 0 )) = (♯‘(𝑚 supp 0 )))
225224eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = 𝑚 → (𝑗 = (♯‘( supp 0 )) ↔ 𝑗 = (♯‘(𝑚 supp 0 ))))
226 eleq1w 2875 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ( = 𝑚 → (𝐻𝑚𝐻))
227225, 226imbi12d 348 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( = 𝑚 → ((𝑗 = (♯‘( supp 0 )) → 𝐻) ↔ (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚𝐻)))
228227rspccva 3573 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻) ∧ 𝑚 ∈ (𝐵m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚𝐻))
229228adantll 713 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ 𝑚 ∈ (𝐵m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚𝐻))
230229imp 410 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ 𝑚 ∈ (𝐵m 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚𝐻)
231230adantllr 718 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ 𝑚 ∈ (𝐵m 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚𝐻)
232231adantlrr 720 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚𝐻)
233232adantrr 716 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → 𝑚𝐻)
234 simplrr 777 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → 𝑧𝐼)
23592ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → 𝑙 ∈ (𝐵m 𝐼))
23691, 235, 234mapfvd 8430 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → (𝑙𝑧) ∈ 𝐵)
23768ad5antr 733 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → ∀𝑎𝐼𝑏𝐵 (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
238 equequ2 2033 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎 = 𝑧 → (𝑥 = 𝑎𝑥 = 𝑧))
239238ifbid 4450 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑧 → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = 𝑧, 𝑏, 0 ))
240239mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑧 → (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )))
241240eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑧 → ((𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻))
242 fveq2 6649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑥 = 𝑧 → (𝑙𝑥) = (𝑙𝑧))
243242eqeq2d 2812 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 = 𝑧 → (𝑏 = (𝑙𝑥) ↔ 𝑏 = (𝑙𝑧)))
244243biimparc 483 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑏 = (𝑙𝑧) ∧ 𝑥 = 𝑧) → 𝑏 = (𝑙𝑥))
245244ifeq1da 4458 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑏 = (𝑙𝑧) → if(𝑥 = 𝑧, 𝑏, 0 ) = if(𝑥 = 𝑧, (𝑙𝑥), 0 ))
246245mpteq2dv 5129 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑏 = (𝑙𝑧) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) = (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))
247246eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = (𝑙𝑧) → ((𝑥𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )) ∈ 𝐻))
248241, 247rspc2va 3585 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧𝐼 ∧ (𝑙𝑧) ∈ 𝐵) ∧ ∀𝑎𝐼𝑏𝐵 (𝑥𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )) ∈ 𝐻)
249234, 236, 237, 248syl21anc 836 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )) ∈ 𝐻)
250 fsuppind.2 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)
251250ralrimivva 3159 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑥𝐻𝑦𝐻 (𝑥f + 𝑦) ∈ 𝐻)
252251ad5antr 733 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → ∀𝑥𝐻𝑦𝐻 (𝑥f + 𝑦) ∈ 𝐻)
253 ovrspc2v 7165 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚𝐻 ∧ (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )) ∈ 𝐻) ∧ ∀𝑥𝐻𝑦𝐻 (𝑥f + 𝑦) ∈ 𝐻) → (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))) ∈ 𝐻)
254233, 249, 252, 253syl21anc 836 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))) ∈ 𝐻)
255223, 254eqeltrd 2893 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 ))))) → 𝑙𝐻)
256255ex 416 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵m 𝐼) ∧ 𝑧𝐼)) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))) → 𝑙𝐻))
257256rexlimdvva 3256 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (∃𝑚 ∈ (𝐵m 𝐼)∃𝑧𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚f + (𝑥𝐼 ↦ if(𝑥 = 𝑧, (𝑙𝑥), 0 )))) → 𝑙𝐻))
258222, 257mpd 15 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) ∧ (𝑙 ∈ (𝐵m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙𝐻)
259258exp32 424 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) → (𝑙 ∈ (𝐵m 𝐼) → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙𝐻)))
260259ralrimiv 3151 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) → ∀𝑙 ∈ (𝐵m 𝐼)((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙𝐻))
261 fvoveq1 7162 . . . . . . . . . . . . . . . . . 18 (𝑙 = → (♯‘(𝑙 supp 0 )) = (♯‘( supp 0 )))
262261eqeq2d 2812 . . . . . . . . . . . . . . . . 17 (𝑙 = → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) ↔ (𝑗 + 1) = (♯‘( supp 0 ))))
263 eleq1w 2875 . . . . . . . . . . . . . . . . 17 (𝑙 = → (𝑙𝐻𝐻))
264262, 263imbi12d 348 . . . . . . . . . . . . . . . 16 (𝑙 = → (((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙𝐻) ↔ ((𝑗 + 1) = (♯‘( supp 0 )) → 𝐻)))
265264cbvralvw 3399 . . . . . . . . . . . . . . 15 (∀𝑙 ∈ (𝐵m 𝐼)((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙𝐻) ↔ ∀ ∈ (𝐵m 𝐼)((𝑗 + 1) = (♯‘( supp 0 )) → 𝐻))
266260, 265sylib 221 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ ∀ ∈ (𝐵m 𝐼)(𝑗 = (♯‘( supp 0 )) → 𝐻)) → ∀ ∈ (𝐵m 𝐼)((𝑗 + 1) = (♯‘( supp 0 )) → 𝐻))
2679, 12, 15, 18, 86, 266nnindd 11649 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ∀ ∈ (𝐵m 𝐼)(𝑛 = (♯‘( supp 0 )) → 𝐻))
268267ralrimiva 3152 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ ℕ ∀ ∈ (𝐵m 𝐼)(𝑛 = (♯‘( supp 0 )) → 𝐻))
269 ralcom 3310 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ ∀ ∈ (𝐵m 𝐼)(𝑛 = (♯‘( supp 0 )) → 𝐻) ↔ ∀ ∈ (𝐵m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘( supp 0 )) → 𝐻))
270268, 269sylib 221 . . . . . . . . . . 11 (𝜑 → ∀ ∈ (𝐵m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘( supp 0 )) → 𝐻))
271 biidd 265 . . . . . . . . . . . . . 14 (𝑛 = (♯‘( supp 0 )) → (𝐻𝐻))
272271ceqsralv 3483 . . . . . . . . . . . . 13 ((♯‘( supp 0 )) ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 = (♯‘( supp 0 )) → 𝐻) ↔ 𝐻))
273272biimpcd 252 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (𝑛 = (♯‘( supp 0 )) → 𝐻) → ((♯‘( supp 0 )) ∈ ℕ → 𝐻))
274273ralimi 3131 . . . . . . . . . . 11 (∀ ∈ (𝐵m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘( supp 0 )) → 𝐻) → ∀ ∈ (𝐵m 𝐼)((♯‘( supp 0 )) ∈ ℕ → 𝐻))
275270, 274syl 17 . . . . . . . . . 10 (𝜑 → ∀ ∈ (𝐵m 𝐼)((♯‘( supp 0 )) ∈ ℕ → 𝐻))
276 fvoveq1 7162 . . . . . . . . . . . . 13 ( = 𝑋 → (♯‘( supp 0 )) = (♯‘(𝑋 supp 0 )))
277276eleq1d 2877 . . . . . . . . . . . 12 ( = 𝑋 → ((♯‘( supp 0 )) ∈ ℕ ↔ (♯‘(𝑋 supp 0 )) ∈ ℕ))
278 eleq1 2880 . . . . . . . . . . . 12 ( = 𝑋 → (𝐻𝑋𝐻))
279277, 278imbi12d 348 . . . . . . . . . . 11 ( = 𝑋 → (((♯‘( supp 0 )) ∈ ℕ → 𝐻) ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋𝐻)))
280279rspcv 3569 . . . . . . . . . 10 (𝑋 ∈ (𝐵m 𝐼) → (∀ ∈ (𝐵m 𝐼)((♯‘( supp 0 )) ∈ ℕ → 𝐻) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋𝐻)))
281275, 280syl5com 31 . . . . . . . . 9 (𝜑 → (𝑋 ∈ (𝐵m 𝐼) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋𝐻)))
282281com23 86 . . . . . . . 8 (𝜑 → ((♯‘(𝑋 supp 0 )) ∈ ℕ → (𝑋 ∈ (𝐵m 𝐼) → 𝑋𝐻)))
283282imp 410 . . . . . . 7 ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋 ∈ (𝐵m 𝐼) → 𝑋𝐻))
2846, 283sylbird 263 . . . . . 6 ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋:𝐼𝐵𝑋𝐻))
285284imp 410 . . . . 5 (((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) ∧ 𝑋:𝐼𝐵) → 𝑋𝐻)
286285an32s 651 . . . 4 (((𝜑𝑋:𝐼𝐵) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋𝐻)
287286adantlr 714 . . 3 ((((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋𝐻)
288 ovex 7172 . . . . 5 (𝑋 supp 0 ) ∈ V
289 hasheq0 13724 . . . . 5 ((𝑋 supp 0 ) ∈ V → ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅))
290288, 289ax-mp 5 . . . 4 ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅)
291 ffn 6491 . . . . . . . 8 (𝑋:𝐼𝐵𝑋 Fn 𝐼)
292291ad2antlr 726 . . . . . . 7 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 Fn 𝐼)
2934ad2antrr 725 . . . . . . 7 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → 𝐼𝑉)
29428a1i 11 . . . . . . 7 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → 0 ∈ V)
295 fnsuppeq0 7845 . . . . . . 7 ((𝑋 Fn 𝐼𝐼𝑉0 ∈ V) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
296292, 293, 294, 295syl3anc 1368 . . . . . 6 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
297296biimpa 480 . . . . 5 ((((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 = (𝐼 × { 0 }))
298 fsuppind.0 . . . . . 6 (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
299298ad3antrrr 729 . . . . 5 ((((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → (𝐼 × { 0 }) ∈ 𝐻)
300297, 299eqeltrd 2893 . . . 4 ((((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋𝐻)
301290, 300sylan2b 596 . . 3 ((((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) = 0) → 𝑋𝐻)
302 simpr 488 . . . . . 6 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 finSupp 0 )
303302fsuppimpd 8828 . . . . 5 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → (𝑋 supp 0 ) ∈ Fin)
304 hashcl 13717 . . . . 5 ((𝑋 supp 0 ) ∈ Fin → (♯‘(𝑋 supp 0 )) ∈ ℕ0)
305303, 304syl 17 . . . 4 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → (♯‘(𝑋 supp 0 )) ∈ ℕ0)
306 elnn0 11891 . . . 4 ((♯‘(𝑋 supp 0 )) ∈ ℕ0 ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
307305, 306sylib 221 . . 3 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
308287, 301, 307mpjaodan 956 . 2 (((𝜑𝑋:𝐼𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋𝐻)
309308anasss 470 1 ((𝜑 ∧ (𝑋:𝐼𝐵𝑋 finSupp 0 )) → 𝑋𝐻)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  ∃!weu 2631  wne 2990  wral 3109  wrex 3110  ∃!wreu 3111  {crab 3113  Vcvv 3444  cdif 3881  c0 4246  ifcif 4428  {csn 4528   class class class wbr 5033  cmpt 5113   × cxp 5521   Fn wfn 6323  wf 6324  cfv 6328  crio 7096  (class class class)co 7139  f cof 7391   supp csupp 7817  m cmap 8393  Fincfn 8496   finSupp cfsupp 8821  0cc0 10530  1c1 10531   + caddc 10533  cn 11629  0cn0 11889  chash 13690  Basecbs 16478  +gcplusg 16560  0gc0g 16708  Grpcgrp 18098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-hash 13691  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101
This theorem is referenced by:  fsuppssind  39446
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