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Theorem fsuppssind 41167
Description: Induction on functions 𝐹:𝐴𝐵 with finite support (see fsuppind 41164) 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 6758 . . . 4 (𝜑 → (𝑋𝑆):𝑆𝐵)
4 fsuppssind.4 . . . . 5 (𝜑𝑋 finSupp 0 )
5 fsuppssind.z . . . . . . 7 0 = (0g𝐺)
65fvexi 6905 . . . . . 6 0 ∈ V
76a1i 11 . . . . 5 (𝜑0 ∈ V)
84, 7fsuppres 9387 . . . 4 (𝜑 → (𝑋𝑆) finSupp 0 )
93, 8jca 512 . . 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 5324 . . . 4 (𝜑𝑆 ∈ V)
1510, 5grpidcl 18849 . . . . . . 7 (𝐺 ∈ Grp → 0𝐵)
1612, 15syl 17 . . . . . 6 (𝜑0𝐵)
17 fconst6g 6780 . . . . . 6 ( 0𝐵 → (𝑆 × { 0 }):𝑆𝐵)
1816, 17syl 17 . . . . 5 (𝜑 → (𝑆 × { 0 }):𝑆𝐵)
19 xpundir 5745 . . . . . . 7 ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 }))
20 undif 4481 . . . . . . . . 9 (𝑆𝐼 ↔ (𝑆 ∪ (𝐼𝑆)) = 𝐼)
212, 20sylib 217 . . . . . . . 8 (𝜑 → (𝑆 ∪ (𝐼𝑆)) = 𝐼)
2221xpeq1d 5705 . . . . . . 7 (𝜑 → ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = (𝐼 × { 0 }))
2319, 22eqtr3id 2786 . . . . . 6 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) = (𝐼 × { 0 }))
24 fsuppssind.0 . . . . . 6 (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
2523, 24eqeltrd 2833 . . . . 5 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
2610fvexi 6905 . . . . . . 7 𝐵 ∈ V
2726a1i 11 . . . . . 6 (𝜑𝐵 ∈ V)
2827, 13, 2fsuppssindlem2 41166 . . . . 5 (𝜑 → ((𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑆 × { 0 }):𝑆𝐵 ∧ ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
2918, 25, 28mpbir2and 711 . . . 4 (𝜑 → (𝑆 × { 0 }) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
30 simplrr 776 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 𝑏𝐵)
3116ad2antrr 724 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 0𝐵)
3230, 31ifcld 4574 . . . . . 6 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → if(𝑠 = 𝑎, 𝑏, 0 ) ∈ 𝐵)
3332fmpttd 7114 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )):𝑆𝐵)
34 fconstmpt 5738 . . . . . . . 8 ((𝐼𝑆) × { 0 }) = (𝑠 ∈ (𝐼𝑆) ↦ 0 )
3534uneq2i 4160 . . . . . . 7 ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 ))
36 eldifn 4127 . . . . . . . . . . . 12 (𝑠 ∈ (𝐼𝑆) → ¬ 𝑠𝑆)
37 eleq1a 2828 . . . . . . . . . . . . . . 15 (𝑎𝑆 → (𝑠 = 𝑎𝑠𝑆))
3837con3dimp 409 . . . . . . . . . . . . . 14 ((𝑎𝑆 ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
3938adantlr 713 . . . . . . . . . . . . 13 (((𝑎𝑆𝑏𝐵) ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
4039adantll 712 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ ¬ 𝑠𝑆) → ¬ 𝑠 = 𝑎)
4136, 40sylan2 593 . . . . . . . . . . 11 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠 ∈ (𝐼𝑆)) → ¬ 𝑠 = 𝑎)
4241iffalsed 4539 . . . . . . . . . 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 4163 . . . . . . . 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 6696 . . . . . . . . 9 (𝑠 ∈ (𝑆 ∪ (𝐼𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
462adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝑆𝐼)
4746, 20sylib 217 . . . . . . . . . 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 2786 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 ))) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5044, 49eqtr3d 2774 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 )) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5135, 50eqtrid 2784 . . . . . 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 2833 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
5426a1i 11 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝐵 ∈ V)
5513adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → 𝐼𝑉)
5654, 55, 46fsuppssindlem2 41166 . . . . 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 711 . . . 4 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
5827, 13, 2fsuppssindlem2 41166 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
5927, 13, 2fsuppssindlem2 41166 . . . . . 6 (𝜑 → (𝑡 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
6058, 59anbi12d 631 . . . . 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 18826 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
6212, 61syl3an1 1163 . . . . . . . . 9 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
63623expb 1120 . . . . . . . 8 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
6463adantlr 713 . . . . . . 7 (((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) ∧ (𝑢𝐵𝑣𝐵)) → (𝑢 + 𝑣) ∈ 𝐵)
65 simprll 777 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑠:𝑆𝐵)
66 simprrl 779 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑡:𝑆𝐵)
6714adantr 481 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑆 ∈ V)
68 inidm 4218 . . . . . . 7 (𝑆𝑆) = 𝑆
6964, 65, 66, 67, 67, 68off 7687 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝑠f + 𝑡):𝑆𝐵)
7065ffnd 6718 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑠 Fn 𝑆)
7166ffnd 6718 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑡 Fn 𝑆)
72 fnconstg 6779 . . . . . . . . . 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 5329 . . . . . . . . . 10 (𝜑 → (𝐼𝑆) ∈ V)
7574adantr 481 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝐼𝑆) ∈ V)
76 disjdif 4471 . . . . . . . . . 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 41060 . . . . . . . 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 7202 . . . . . . . . . . . 12 (( 0 ∈ V ∧ 𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
817, 80sylan 580 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
8210, 11, 5grplid 18851 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 0𝐵) → ( 0 + 0 ) = 0 )
8312, 16, 82syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ( 0 + 0 ) = 0 )
8483adantr 481 . . . . . . . . . . . 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 2775 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → ( 0 + 0 ) = (((𝐼𝑆) × { 0 })‘𝑗))
8874, 79, 79, 79, 81, 81, 87offveq 7693 . . . . . . . . . 10 (𝜑 → (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 })) = ((𝐼𝑆) × { 0 }))
8988uneq2d 4163 . . . . . . . . 9 (𝜑 → ((𝑠f + 𝑡) ∪ (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 }))) = ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })))
9089adantr 481 . . . . . . . 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 2772 . . . . . . 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 7598 . . . . . . . . 9 ((𝜑 ∧ ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9493adantrrl 722 . . . . . . . 8 ((𝜑 ∧ ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻 ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9594adantrll 720 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠 ∪ ((𝐼𝑆) × { 0 })) ∘f + (𝑡 ∪ ((𝐼𝑆) × { 0 }))) ∈ 𝐻)
9691, 95eqeltrrd 2834 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
9727, 13, 2fsuppssindlem2 41166 . . . . . . 7 (𝜑 → ((𝑠f + 𝑡) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ ((𝑠f + 𝑡):𝑆𝐵 ∧ ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
9897adantr 481 . . . . . 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 711 . . . . 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 41164 . . 3 ((𝜑 ∧ ((𝑋𝑆):𝑆𝐵 ∧ (𝑋𝑆) finSupp 0 )) → (𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
1029, 101mpdan 685 . 2 (𝜑 → (𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻})
10327, 14elmapd 8833 . . . . 5 (𝜑 → ((𝑋𝑆) ∈ (𝐵m 𝑆) ↔ (𝑋𝑆):𝑆𝐵))
1043, 103mpbird 256 . . . 4 (𝜑 → (𝑋𝑆) ∈ (𝐵m 𝑆))
105 fveq1 6890 . . . . . . . 8 (𝑓 = (𝑋𝑆) → (𝑓𝑖) = ((𝑋𝑆)‘𝑖))
106105ifeq1d 4547 . . . . . . 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 2818 . . . . 5 (𝑓 = (𝑋𝑆) → ((𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻 ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
109108elrab3 3684 . . . 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 41165 . . . 4 (𝜑𝑋 = (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )))
113112eleq1d 2818 . . 3 (𝜑 → (𝑋𝐻 ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
114110, 113bitr4d 281 . 2 (𝜑 → ((𝑋𝑆) ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ 𝑋𝐻))
115102, 114mpbid 231 1 (𝜑𝑋𝐻)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {crab 3432  Vcvv 3474  cdif 3945  cun 3946  cin 3947  wss 3948  c0 4322  ifcif 4528  {csn 4628   class class class wbr 5148  cmpt 5231   × cxp 5674  cres 5678   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7408  f cof 7667   supp csupp 8145  m cmap 8819   finSupp cfsupp 9360  Basecbs 17143  +gcplusg 17196  0gc0g 17384  Grpcgrp 18818
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7669  df-om 7855  df-1st 7974  df-2nd 7975  df-supp 8146  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-oadd 8469  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fsupp 9361  df-dju 9895  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-hash 14290  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821
This theorem is referenced by:  mhpind  41168
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