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Theorem fsuppssind 42580
Description: Induction on functions 𝐹:𝐴𝐵 with finite support (see fsuppind 42577) whose supports are subsets of 𝑆. (Contributed by SN, 15-Jun-2024.)
Hypotheses
Ref Expression
fsuppssind.b 𝐵 = (Base‘𝐺)
fsuppssind.z 0 = (0g𝐺)
fsuppssind.p + = (+g𝐺)
fsuppssind.g (𝜑𝐺 ∈ Grp)
fsuppssind.v (𝜑𝐼𝑉)
fsuppssind.s (𝜑𝑆𝐼)
fsuppssind.0 (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
fsuppssind.1 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
fsuppssind.2 ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)
fsuppssind.3 (𝜑𝑋:𝐼𝐵)
fsuppssind.4 (𝜑𝑋 finSupp 0 )
fsuppssind.5 (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)
Assertion
Ref Expression
fsuppssind (𝜑𝑋𝐻)
Distinct variable groups:   𝐵,𝑎,𝑏,𝑠   0 ,𝑎,𝑏,𝑠   𝑥, 0 ,𝑦   + ,𝑠,𝑥,𝑦   𝜑,𝑎,𝑏,𝑠   𝜑,𝑥,𝑦   𝐼,𝑎,𝑏,𝑠   𝑥,𝐼,𝑦,𝑠   𝑆,𝑎,𝑏   𝑥,𝑆,𝑦,𝑠   𝐻,𝑎,𝑏,𝑠   𝑥,𝐻,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   + (𝑎,𝑏)   𝐺(𝑥,𝑦,𝑠,𝑎,𝑏)   𝑉(𝑥,𝑦,𝑠,𝑎,𝑏)   𝑋(𝑥,𝑦,𝑠,𝑎,𝑏)

Proof of Theorem fsuppssind
Dummy variables 𝑓 𝑡 𝑢 𝑣 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsuppssind.3 . . . . 5 (𝜑𝑋:𝐼𝐵)
2 fsuppssind.s . . . . 5 (𝜑𝑆𝐼)
31, 2fssresd 6776 . . . 4 (𝜑 → (𝑋𝑆):𝑆𝐵)
4 fsuppssind.4 . . . . 5 (𝜑𝑋 finSupp 0 )
5 fsuppssind.z . . . . . . 7 0 = (0g𝐺)
65fvexi 6921 . . . . . 6 0 ∈ V
76a1i 11 . . . . 5 (𝜑0 ∈ V)
84, 7fsuppres 9431 . . . 4 (𝜑 → (𝑋𝑆) finSupp 0 )
93, 8jca 511 . . 3 (𝜑 → ((𝑋𝑆):𝑆𝐵 ∧ (𝑋𝑆) finSupp 0 ))
10 fsuppssind.b . . . 4 𝐵 = (Base‘𝐺)
11 fsuppssind.p . . . 4 + = (+g𝐺)
12 fsuppssind.g . . . 4 (𝜑𝐺 ∈ Grp)
13 fsuppssind.v . . . . 5 (𝜑𝐼𝑉)
1413, 2ssexd 5330 . . . 4 (𝜑𝑆 ∈ V)
1510, 5grpidcl 18996 . . . . . . 7 (𝐺 ∈ Grp → 0𝐵)
1612, 15syl 17 . . . . . 6 (𝜑0𝐵)
17 fconst6g 6798 . . . . . 6 ( 0𝐵 → (𝑆 × { 0 }):𝑆𝐵)
1816, 17syl 17 . . . . 5 (𝜑 → (𝑆 × { 0 }):𝑆𝐵)
19 xpundir 5758 . . . . . . 7 ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 }))
20 undif 4488 . . . . . . . . 9 (𝑆𝐼 ↔ (𝑆 ∪ (𝐼𝑆)) = 𝐼)
212, 20sylib 218 . . . . . . . 8 (𝜑 → (𝑆 ∪ (𝐼𝑆)) = 𝐼)
2221xpeq1d 5718 . . . . . . 7 (𝜑 → ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = (𝐼 × { 0 }))
2319, 22eqtr3id 2789 . . . . . 6 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) = (𝐼 × { 0 }))
24 fsuppssind.0 . . . . . 6 (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
2523, 24eqeltrd 2839 . . . . 5 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
2610fvexi 6921 . . . . . . 7 𝐵 ∈ V
2726a1i 11 . . . . . 6 (𝜑𝐵 ∈ V)
2827, 13, 2fsuppssindlem2 42579 . . . . 5 (𝜑 → ((𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑆 × { 0 }):𝑆𝐵 ∧ ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
2918, 25, 28mpbir2and 713 . . . 4 (𝜑 → (𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
30 simplrr 778 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 𝑏𝐵)
3116ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 0𝐵)
3230, 31ifcld 4577 . . . . . 6 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → if(𝑠 = 𝑎, 𝑏, 0 ) ∈ 𝐵)
3332fmpttd 7135 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )):𝑆𝐵)
34 fconstmpt 5751 . . . . . . . 8 ((𝐼𝑆) × { 0 }) = (𝑠 ∈ (𝐼𝑆) ↦ 0 )
3534uneq2i 4175 . . . . . . 7 ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 ))
36 eldifn 4142 . . . . . . . . . . . 12 (𝑠 ∈ (𝐼𝑆) → ¬ 𝑠𝑆)
37 eleq1a 2834 . . . . . . . . . . . . . . 15 (𝑎𝑆 → (𝑠 = 𝑎𝑠𝑆))
3837con3dimp 408 . . . . . . . . . . . . . 14 ((𝑎𝑆 ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
3938adantlr 715 . . . . . . . . . . . . 13 (((𝑎𝑆𝑏𝐵) ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
4039adantll 714 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
4136, 40sylan2 593 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠 ∈ (𝐼𝑆)) → ¬ 𝑠 = 𝑎)
4241iffalsed 4542 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠 ∈ (𝐼𝑆)) → if(𝑠 = 𝑎, 𝑏, 0 ) = 0 )
4342mpteq2dva 5248 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠 ∈ (𝐼𝑆) ↦ 0 ))
4443uneq2d 4178 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 ))) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 )))
45 mptun 6715 . . . . . . . . 9 (𝑠 ∈ (𝑆 ∪ (𝐼𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
462adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝑆𝐼)
4746, 20sylib 218 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑆 ∪ (𝐼𝑆)) = 𝐼)
4847mpteq1d 5243 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠 ∈ (𝑆 ∪ (𝐼𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
4945, 48eqtr3id 2789 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 ))) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5044, 49eqtr3d 2777 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 )) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5135, 50eqtrid 2787 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
52 fsuppssind.1 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
5351, 52eqeltrd 2839 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
5426a1i 11 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝐵 ∈ V)
5513adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝐼𝑉)
5654, 55, 46fsuppssindlem2 42579 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )):𝑆𝐵 ∧ ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
5733, 53, 56mpbir2and 713 . . . 4 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
5827, 13, 2fsuppssindlem2 42579 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
5927, 13, 2fsuppssindlem2 42579 . . . . . 6 (𝜑 → (𝑡 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
6058, 59anbi12d 632 . . . . 5 (𝜑 → ((𝑠 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ∧ 𝑡 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻}) ↔ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))))
6110, 11grpcl 18972 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
6212, 61syl3an1 1162 . . . . . . . . 9 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
63623expb 1119 . . . . . . . 8 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
6463adantlr 715 . . . . . . 7 (((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
65 simprll 779 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑠:𝑆𝐵)
66 simprrl 781 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑡:𝑆𝐵)
6714adantr 480 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑆 ∈ V)
68 inidm 4235 . . . . . . 7 (𝑆𝑆) = 𝑆
6964, 65, 66, 67, 67, 68off 7715 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝑠f + 𝑡):𝑆𝐵)
7065ffnd 6738 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑠 Fn 𝑆)
7166ffnd 6738 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑡 Fn 𝑆)
72 fnconstg 6797 . . . . . . . . . 10 ( 0 ∈ V → ((𝐼𝑆) × { 0 }) Fn (𝐼𝑆))
736, 72mp1i 13 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝐼𝑆) × { 0 }) Fn (𝐼𝑆))
7413difexd 5337 . . . . . . . . . 10 (𝜑 → (𝐼𝑆) ∈ V)
7574adantr 480 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝐼𝑆) ∈ V)
76 disjdif 4478 . . . . . . . . . 10 (𝑆 ∩ (𝐼𝑆)) = ∅
7776a1i 11 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝑆 ∩ (𝐼𝑆)) = ∅)
7870, 71, 73, 73, 67, 75, 77ofun 42256 . . . . . . . 8 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) = ((𝑠f + 𝑡) ∪ (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 }))))
796, 72mp1i 13 . . . . . . . . . . 11 (𝜑 → ((𝐼𝑆) × { 0 }) Fn (𝐼𝑆))
80 fvconst2g 7222 . . . . . . . . . . . 12 (( 0 ∈ V ∧ 𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
817, 80sylan 580 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
8210, 11, 5grplid 18998 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
8312, 16, 82syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ( 0 + 0 ) = 0 )
8483adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝐼𝑆)) → ( 0 + 0 ) = 0 )
856a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝐼𝑆)) → 0 ∈ V)
8685, 80sylancom 588 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
8784, 86eqtr4d 2778 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → ( 0 + 0 ) = (((𝐼𝑆) × { 0 })‘𝑗))
8874, 79, 79, 79, 81, 81, 87offveq 7723 . . . . . . . . . 10 (𝜑 → (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 })) = ((𝐼𝑆) × { 0 }))
8988uneq2d 4178 . . . . . . . . 9 (𝜑 → ((𝑠f + 𝑡) ∪ (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 }))) = ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })))
9089adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠f + 𝑡) ∪ (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 }))) = ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })))
9178, 90eqtrd 2775 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) = ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })))
92 fsuppssind.2 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐻𝑦𝐻)) → (𝑥f + 𝑦) ∈ 𝐻)
9392caovclg 7625 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9493adantrrl 724 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9594adantrll 722 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9691, 95eqeltrrd 2840 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
9727, 13, 2fsuppssindlem2 42579 . . . . . . 7 (𝜑 → ((𝑠f + 𝑡) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠f + 𝑡):𝑆𝐵 ∧ ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
9897adantr 480 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠f + 𝑡) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠f + 𝑡):𝑆𝐵 ∧ ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
9969, 96, 98mpbir2and 713 . . . . 5 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝑠f + 𝑡) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
10060, 99sylbida 592 . . . 4 ((𝜑 ∧ (𝑠 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ∧ 𝑡 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})) → (𝑠f + 𝑡) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
10110, 5, 11, 12, 14, 29, 57, 100fsuppind 42577 . . 3 ((𝜑 ∧ ((𝑋𝑆):𝑆𝐵 ∧ (𝑋𝑆) finSupp 0 )) → (𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
1029, 101mpdan 687 . 2 (𝜑 → (𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
10327, 14elmapd 8879 . . . . 5 (𝜑 → ((𝑋𝑆) ∈ (𝐵m 𝑆) ↔ (𝑋𝑆):𝑆𝐵))
1043, 103mpbird 257 . . . 4 (𝜑 → (𝑋𝑆) ∈ (𝐵m 𝑆))
105 fveq1 6906 . . . . . . . 8 (𝑓 = (𝑋𝑆) → (𝑓𝑖) = ((𝑋𝑆)‘𝑖))
106105ifeq1d 4550 . . . . . . 7 (𝑓 = (𝑋𝑆) → if(𝑖𝑆, (𝑓𝑖), 0 ) = if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 ))
107106mpteq2dv 5250 . . . . . 6 (𝑓 = (𝑋𝑆) → (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) = (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )))
108107eleq1d 2824 . . . . 5 (𝑓 = (𝑋𝑆) → ((𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻 ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
109108elrab3 3696 . . . 4 ((𝑋𝑆) ∈ (𝐵m 𝑆) → ((𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
110104, 109syl 17 . . 3 (𝜑 → ((𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
111 fsuppssind.5 . . . . 5 (𝜑 → (𝑋 supp 0 ) ⊆ 𝑆)
1127, 13, 1, 111fsuppssindlem1 42578 . . . 4 (𝜑𝑋 = (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )))
113112eleq1d 2824 . . 3 (𝜑 → (𝑋𝐻 ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
114110, 113bitr4d 282 . 2 (𝜑 → ((𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ 𝑋𝐻))
115102, 114mpbid 232 1 (𝜑𝑋𝐻)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  ifcif 4531  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  cres 5691   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695   supp csupp 8184  m cmap 8865   finSupp cfsupp 9399  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Grpcgrp 18964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967
This theorem is referenced by:  mhpind  42581
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