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Theorem fsuppssind 39446
Description: Induction on functions 𝐹:𝐴𝐵 with finite support (see fsuppind 39443) 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 6523 . . . 4 (𝜑 → (𝑋𝑆):𝑆𝐵)
4 fsuppssind.4 . . . . 5 (𝜑𝑋 finSupp 0 )
5 fsuppssind.z . . . . . . 7 0 = (0g𝐺)
65fvexi 6663 . . . . . 6 0 ∈ V
76a1i 11 . . . . 5 (𝜑0 ∈ V)
84, 7fsuppres 8846 . . . 4 (𝜑 → (𝑋𝑆) finSupp 0 )
93, 8jca 515 . . 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 5195 . . . 4 (𝜑𝑆 ∈ V)
1510, 5grpidcl 18126 . . . . . . 7 (𝐺 ∈ Grp → 0𝐵)
1612, 15syl 17 . . . . . 6 (𝜑0𝐵)
17 fconst6g 6546 . . . . . 6 ( 0𝐵 → (𝑆 × { 0 }):𝑆𝐵)
1816, 17syl 17 . . . . 5 (𝜑 → (𝑆 × { 0 }):𝑆𝐵)
19 xpundir 5589 . . . . . . 7 ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 }))
20 undif 4391 . . . . . . . . 9 (𝑆𝐼 ↔ (𝑆 ∪ (𝐼𝑆)) = 𝐼)
212, 20sylib 221 . . . . . . . 8 (𝜑 → (𝑆 ∪ (𝐼𝑆)) = 𝐼)
2221xpeq1d 5552 . . . . . . 7 (𝜑 → ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = (𝐼 × { 0 }))
2319, 22syl5eqr 2850 . . . . . 6 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) = (𝐼 × { 0 }))
24 fsuppssind.0 . . . . . 6 (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
2523, 24eqeltrd 2893 . . . . 5 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
2610fvexi 6663 . . . . . . 7 𝐵 ∈ V
2726a1i 11 . . . . . 6 (𝜑𝐵 ∈ V)
2827, 13, 2fsuppssindlem2 39445 . . . . 5 (𝜑 → ((𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑆 × { 0 }):𝑆𝐵 ∧ ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
2918, 25, 28mpbir2and 712 . . . 4 (𝜑 → (𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
30 simplrr 777 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 𝑏𝐵)
3116ad2antrr 725 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 0𝐵)
3230, 31ifcld 4473 . . . . . 6 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → if(𝑠 = 𝑎, 𝑏, 0 ) ∈ 𝐵)
3332fmpttd 6860 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )):𝑆𝐵)
34 fconstmpt 5582 . . . . . . . 8 ((𝐼𝑆) × { 0 }) = (𝑠 ∈ (𝐼𝑆) ↦ 0 )
3534uneq2i 4090 . . . . . . 7 ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 ))
36 eldifn 4058 . . . . . . . . . . . 12 (𝑠 ∈ (𝐼𝑆) → ¬ 𝑠𝑆)
37 eleq1a 2888 . . . . . . . . . . . . . . 15 (𝑎𝑆 → (𝑠 = 𝑎𝑠𝑆))
3837con3dimp 412 . . . . . . . . . . . . . 14 ((𝑎𝑆 ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
3938adantlr 714 . . . . . . . . . . . . 13 (((𝑎𝑆𝑏𝐵) ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
4039adantll 713 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
4136, 40sylan2 595 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠 ∈ (𝐼𝑆)) → ¬ 𝑠 = 𝑎)
4241iffalsed 4439 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠 ∈ (𝐼𝑆)) → if(𝑠 = 𝑎, 𝑏, 0 ) = 0 )
4342mpteq2dva 5128 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠 ∈ (𝐼𝑆) ↦ 0 ))
4443uneq2d 4093 . . . . . . . 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 6470 . . . . . . . . 9 (𝑠 ∈ (𝑆 ∪ (𝐼𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
462adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝑆𝐼)
4746, 20sylib 221 . . . . . . . . . 10 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑆 ∪ (𝐼𝑆)) = 𝐼)
4847mpteq1d 5122 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠 ∈ (𝑆 ∪ (𝐼𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
4945, 48syl5eqr 2850 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 ))) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5044, 49eqtr3d 2838 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 )) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5135, 50syl5eq 2848 . . . . . 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 2893 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
5426a1i 11 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝐵 ∈ V)
5513adantr 484 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝐼𝑉)
5654, 55, 46fsuppssindlem2 39445 . . . . 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 712 . . . 4 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
5827, 13, 2fsuppssindlem2 39445 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
5927, 13, 2fsuppssindlem2 39445 . . . . . 6 (𝜑 → (𝑡 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
6058, 59anbi12d 633 . . . . 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 18106 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
6212, 61syl3an1 1160 . . . . . . . . 9 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
63623expb 1117 . . . . . . . 8 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
6463adantlr 714 . . . . . . 7 (((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
65 simprll 778 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑠:𝑆𝐵)
66 simprrl 780 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑡:𝑆𝐵)
6714adantr 484 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑆 ∈ V)
68 inidm 4148 . . . . . . 7 (𝑆𝑆) = 𝑆
6964, 65, 66, 67, 67, 68off 7408 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝑠f + 𝑡):𝑆𝐵)
7065ffnd 6492 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑠 Fn 𝑆)
7166ffnd 6492 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑡 Fn 𝑆)
72 fnconstg 6545 . . . . . . . . . 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 39391 . . . . . . . . . 10 (𝜑 → (𝐼𝑆) ∈ V)
7574adantr 484 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝐼𝑆) ∈ V)
76 disjdif 4382 . . . . . . . . . 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 39403 . . . . . . . 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 6945 . . . . . . . . . . . 12 (( 0 ∈ V ∧ 𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
817, 80sylan 583 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
8210, 11, 5grplid 18128 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
8312, 16, 82syl2anc 587 . . . . . . . . . . . . 13 (𝜑 → ( 0 + 0 ) = 0 )
8483adantr 484 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝐼𝑆)) → ( 0 + 0 ) = 0 )
856a1i 11 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (𝐼𝑆)) → 0 ∈ V)
8685, 80sylancom 591 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
8784, 86eqtr4d 2839 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → ( 0 + 0 ) = (((𝐼𝑆) × { 0 })‘𝑗))
8874, 79, 79, 79, 81, 81, 87offveq 7414 . . . . . . . . . 10 (𝜑 → (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 })) = ((𝐼𝑆) × { 0 }))
8988uneq2d 4093 . . . . . . . . 9 (𝜑 → ((𝑠f + 𝑡) ∪ (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 }))) = ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })))
9089adantr 484 . . . . . . . 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 2836 . . . . . . 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 7324 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9493adantrrl 723 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9594adantrll 721 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9691, 95eqeltrrd 2894 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
9727, 13, 2fsuppssindlem2 39445 . . . . . . 7 (𝜑 → ((𝑠f + 𝑡) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠f + 𝑡):𝑆𝐵 ∧ ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
9897adantr 484 . . . . . 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 712 . . . . 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, 99sylibda 39378 . . . 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 39443 . . 3 ((𝜑 ∧ ((𝑋𝑆):𝑆𝐵 ∧ (𝑋𝑆) finSupp 0 )) → (𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
1029, 101mpdan 686 . 2 (𝜑 → (𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
10327, 14elmapd 8407 . . . . 5 (𝜑 → ((𝑋𝑆) ∈ (𝐵m 𝑆) ↔ (𝑋𝑆):𝑆𝐵))
1043, 103mpbird 260 . . . 4 (𝜑 → (𝑋𝑆) ∈ (𝐵m 𝑆))
105 fveq1 6648 . . . . . . . 8 (𝑓 = (𝑋𝑆) → (𝑓𝑖) = ((𝑋𝑆)‘𝑖))
106105ifeq1d 4446 . . . . . . 7 (𝑓 = (𝑋𝑆) → if(𝑖𝑆, (𝑓𝑖), 0 ) = if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 ))
107106mpteq2dv 5129 . . . . . 6 (𝑓 = (𝑋𝑆) → (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) = (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )))
108107eleq1d 2877 . . . . 5 (𝑓 = (𝑋𝑆) → ((𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻 ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
109108elrab3 3632 . . . 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 39444 . . . 4 (𝜑𝑋 = (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )))
113112eleq1d 2877 . . 3 (𝜑 → (𝑋𝐻 ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
114110, 113bitr4d 285 . 2 (𝜑 → ((𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ 𝑋𝐻))
115102, 114mpbid 235 1 (𝜑𝑋𝐻)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  ifcif 4428  {csn 4528   class class class wbr 5033  cmpt 5113   × cxp 5521  cres 5525   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  f cof 7391   supp csupp 7817  m cmap 8393   finSupp cfsupp 8821  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: (None)
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