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Theorem fsuppssind 42608
Description: Induction on functions 𝐹:𝐴𝐵 with finite support (see fsuppind 42605) 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 6774 . . . 4 (𝜑 → (𝑋𝑆):𝑆𝐵)
4 fsuppssind.4 . . . . 5 (𝜑𝑋 finSupp 0 )
5 fsuppssind.z . . . . . . 7 0 = (0g𝐺)
65fvexi 6919 . . . . . 6 0 ∈ V
76a1i 11 . . . . 5 (𝜑0 ∈ V)
84, 7fsuppres 9434 . . . 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 5323 . . . 4 (𝜑𝑆 ∈ V)
1510, 5grpidcl 18984 . . . . . . 7 (𝐺 ∈ Grp → 0𝐵)
1612, 15syl 17 . . . . . 6 (𝜑0𝐵)
17 fconst6g 6796 . . . . . 6 ( 0𝐵 → (𝑆 × { 0 }):𝑆𝐵)
1816, 17syl 17 . . . . 5 (𝜑 → (𝑆 × { 0 }):𝑆𝐵)
19 xpundir 5754 . . . . . . 7 ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 }))
20 undif 4481 . . . . . . . . 9 (𝑆𝐼 ↔ (𝑆 ∪ (𝐼𝑆)) = 𝐼)
212, 20sylib 218 . . . . . . . 8 (𝜑 → (𝑆 ∪ (𝐼𝑆)) = 𝐼)
2221xpeq1d 5713 . . . . . . 7 (𝜑 → ((𝑆 ∪ (𝐼𝑆)) × { 0 }) = (𝐼 × { 0 }))
2319, 22eqtr3id 2790 . . . . . 6 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) = (𝐼 × { 0 }))
24 fsuppssind.0 . . . . . 6 (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
2523, 24eqeltrd 2840 . . . . 5 (𝜑 → ((𝑆 × { 0 }) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
2610fvexi 6919 . . . . . . 7 𝐵 ∈ V
2726a1i 11 . . . . . 6 (𝜑𝐵 ∈ V)
2827, 13, 2fsuppssindlem2 42607 . . . . 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 777 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 𝑏𝐵)
3116ad2antrr 726 . . . . . . 7 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → 0𝐵)
3230, 31ifcld 4571 . . . . . 6 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠𝑆) → if(𝑠 = 𝑎, 𝑏, 0 ) ∈ 𝐵)
3332fmpttd 7134 . . . . 5 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )):𝑆𝐵)
34 fconstmpt 5746 . . . . . . . 8 ((𝐼𝑆) × { 0 }) = (𝑠 ∈ (𝐼𝑆) ↦ 0 )
3534uneq2i 4164 . . . . . . 7 ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ ((𝐼𝑆) × { 0 })) = ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 ))
36 eldifn 4131 . . . . . . . . . . . 12 (𝑠 ∈ (𝐼𝑆) → ¬ 𝑠𝑆)
37 eleq1a 2835 . . . . . . . . . . . . . . 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 4535 . . . . . . . . . 10 (((𝜑 ∧ (𝑎𝑆𝑏𝐵)) ∧ 𝑠 ∈ (𝐼𝑆)) → if(𝑠 = 𝑎, 𝑏, 0 ) = 0 )
4342mpteq2dva 5241 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠 ∈ (𝐼𝑆) ↦ 0 ))
4443uneq2d 4167 . . . . . . . 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 6713 . . . . . . . . 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 5236 . . . . . . . . 9 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → (𝑠 ∈ (𝑆 ∪ (𝐼𝑆)) ↦ if(𝑠 = 𝑎, 𝑏, 0 )) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
4945, 48eqtr3id 2790 . . . . . . . 8 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ if(𝑠 = 𝑎, 𝑏, 0 ))) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5044, 49eqtr3d 2778 . . . . . . 7 ((𝜑 ∧ (𝑎𝑆𝑏𝐵)) → ((𝑠𝑆 ↦ if(𝑠 = 𝑎, 𝑏, 0 )) ∪ (𝑠 ∈ (𝐼𝑆) ↦ 0 )) = (𝑠𝐼 ↦ if(𝑠 = 𝑎, 𝑏, 0 )))
5135, 50eqtrid 2788 . . . . . 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 2840 . . . . 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 42607 . . . . 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 42607 . . . . . 6 (𝜑 → (𝑠 ∈ {𝑓 ∈ (𝐵m 𝑆) ∣ (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻} ↔ (𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)))
5927, 13, 2fsuppssindlem2 42607 . . . . . 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 18960 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
6212, 61syl3an1 1163 . . . . . . . . 9 ((𝜑𝑢𝐵𝑣𝐵) → (𝑢 + 𝑣) ∈ 𝐵)
63623expb 1120 . . . . . . . 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 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 480 . . . . . . 7 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑆 ∈ V)
68 inidm 4226 . . . . . . 7 (𝑆𝑆) = 𝑆
6964, 65, 66, 67, 67, 68off 7716 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → (𝑠f + 𝑡):𝑆𝐵)
7065ffnd 6736 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑠 Fn 𝑆)
7166ffnd 6736 . . . . . . . . 9 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → 𝑡 Fn 𝑆)
72 fnconstg 6795 . . . . . . . . . 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 5330 . . . . . . . . . 10 (𝜑 → (𝐼𝑆) ∈ V)
7574adantr 480 . . . . . . . . 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 42277 . . . . . . . 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 7223 . . . . . . . . . . . 12 (( 0 ∈ V ∧ 𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
817, 80sylan 580 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → (((𝐼𝑆) × { 0 })‘𝑗) = 0 )
8210, 11, 5grplid 18986 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . 11 ((𝜑𝑗 ∈ (𝐼𝑆)) → ( 0 + 0 ) = (((𝐼𝑆) × { 0 })‘𝑗))
8874, 79, 79, 79, 81, 81, 87offveq 7724 . . . . . . . . . 10 (𝜑 → (((𝐼𝑆) × { 0 }) ∘f + ((𝐼𝑆) × { 0 })) = ((𝐼𝑆) × { 0 }))
8988uneq2d 4167 . . . . . . . . 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 2776 . . . . . . 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 7626 . . . . . . . . 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 2841 . . . . . 6 ((𝜑 ∧ ((𝑠:𝑆𝐵 ∧ (𝑠 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻) ∧ (𝑡:𝑆𝐵 ∧ (𝑡 ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻))) → ((𝑠f + 𝑡) ∪ ((𝐼𝑆) × { 0 })) ∈ 𝐻)
9727, 13, 2fsuppssindlem2 42607 . . . . . . 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 42605 . . 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 8881 . . . . 5 (𝜑 → ((𝑋𝑆) ∈ (𝐵m 𝑆) ↔ (𝑋𝑆):𝑆𝐵))
1043, 103mpbird 257 . . . 4 (𝜑 → (𝑋𝑆) ∈ (𝐵m 𝑆))
105 fveq1 6904 . . . . . . . 8 (𝑓 = (𝑋𝑆) → (𝑓𝑖) = ((𝑋𝑆)‘𝑖))
106105ifeq1d 4544 . . . . . . 7 (𝑓 = (𝑋𝑆) → if(𝑖𝑆, (𝑓𝑖), 0 ) = if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 ))
107106mpteq2dv 5243 . . . . . 6 (𝑓 = (𝑋𝑆) → (𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) = (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )))
108107eleq1d 2825 . . . . 5 (𝑓 = (𝑋𝑆) → ((𝑖𝐼 ↦ if(𝑖𝑆, (𝑓𝑖), 0 )) ∈ 𝐻 ↔ (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )) ∈ 𝐻))
109108elrab3 3692 . . . 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 42606 . . . 4 (𝜑𝑋 = (𝑖𝐼 ↦ if(𝑖𝑆, ((𝑋𝑆)‘𝑖), 0 )))
113112eleq1d 2825 . . 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 1539  wcel 2107  {crab 3435  Vcvv 3479  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  ifcif 4524  {csn 4625   class class class wbr 5142  cmpt 5224   × cxp 5682  cres 5686   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  f cof 7696   supp csupp 8186  m cmap 8867   finSupp cfsupp 9402  Basecbs 17248  +gcplusg 17298  0gc0g 17485  Grpcgrp 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-hash 14371  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955
This theorem is referenced by:  mhpind  42609
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