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Theorem tsmsxplem1 22003
Description: Lemma for tsmsxp 22005. (Contributed by Mario Carneiro, 21-Sep-2015.)
Hypotheses
Ref Expression
tsmsxp.b 𝐵 = (Base‘𝐺)
tsmsxp.g (𝜑𝐺 ∈ CMnd)
tsmsxp.2 (𝜑𝐺 ∈ TopGrp)
tsmsxp.a (𝜑𝐴𝑉)
tsmsxp.c (𝜑𝐶𝑊)
tsmsxp.f (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)
tsmsxp.h (𝜑𝐻:𝐴𝐵)
tsmsxp.1 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
tsmsxp.j 𝐽 = (TopOpen‘𝐺)
tsmsxp.z 0 = (0g𝐺)
tsmsxp.p + = (+g𝐺)
tsmsxp.m = (-g𝐺)
tsmsxp.l (𝜑𝐿𝐽)
tsmsxp.3 (𝜑0𝐿)
tsmsxp.k (𝜑𝐾 ∈ (𝒫 𝐴 ∩ Fin))
tsmsxp.ks (𝜑 → dom 𝐷𝐾)
tsmsxp.d (𝜑𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
Assertion
Ref Expression
tsmsxplem1 (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷𝑛 ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
Distinct variable groups:   0 ,𝑘   𝑗,𝑘,𝑛,𝑥,𝐺   𝐵,𝑘   𝐷,𝑗,𝑘,𝑛,𝑥   𝑗,𝐿,𝑛,𝑥   𝐴,𝑗,𝑘,𝑛   𝑗,𝐾,𝑘,𝑛,𝑥   𝑗,𝐻,𝑘,𝑛,𝑥   ,𝑗,𝑛,𝑥   𝐶,𝑗,𝑘,𝑛   𝑗,𝐹,𝑘,𝑛,𝑥   𝜑,𝑗,𝑘,𝑛
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥,𝑗,𝑛)   𝐶(𝑥)   + (𝑥,𝑗,𝑘,𝑛)   𝐽(𝑥,𝑗,𝑘,𝑛)   𝐿(𝑘)   (𝑘)   𝑉(𝑥,𝑗,𝑘,𝑛)   𝑊(𝑥,𝑗,𝑘,𝑛)   0 (𝑥,𝑗,𝑛)

Proof of Theorem tsmsxplem1
Dummy variables 𝑔 𝑦 𝑧 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tsmsxp.k . . . 4 (𝜑𝐾 ∈ (𝒫 𝐴 ∩ Fin))
2 elfpw 8309 . . . . 5 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝐾𝐴𝐾 ∈ Fin))
32simprbi 479 . . . 4 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) → 𝐾 ∈ Fin)
41, 3syl 17 . . 3 (𝜑𝐾 ∈ Fin)
52simplbi 475 . . . . . . 7 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) → 𝐾𝐴)
61, 5syl 17 . . . . . 6 (𝜑𝐾𝐴)
76sselda 3636 . . . . 5 ((𝜑𝑗𝐾) → 𝑗𝐴)
8 tsmsxp.b . . . . . 6 𝐵 = (Base‘𝐺)
9 tsmsxp.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
10 eqid 2651 . . . . . 6 (𝒫 𝐶 ∩ Fin) = (𝒫 𝐶 ∩ Fin)
11 tsmsxp.g . . . . . . 7 (𝜑𝐺 ∈ CMnd)
1211adantr 480 . . . . . 6 ((𝜑𝑗𝐴) → 𝐺 ∈ CMnd)
13 tsmsxp.2 . . . . . . . 8 (𝜑𝐺 ∈ TopGrp)
14 tgptps 21931 . . . . . . . 8 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
1513, 14syl 17 . . . . . . 7 (𝜑𝐺 ∈ TopSp)
1615adantr 480 . . . . . 6 ((𝜑𝑗𝐴) → 𝐺 ∈ TopSp)
17 tsmsxp.c . . . . . . 7 (𝜑𝐶𝑊)
1817adantr 480 . . . . . 6 ((𝜑𝑗𝐴) → 𝐶𝑊)
19 tsmsxp.f . . . . . . . . 9 (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)
20 fovrn 6846 . . . . . . . . 9 ((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
2119, 20syl3an1 1399 . . . . . . . 8 ((𝜑𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
22213expa 1284 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
23 eqid 2651 . . . . . . 7 (𝑘𝐶 ↦ (𝑗𝐹𝑘)) = (𝑘𝐶 ↦ (𝑗𝐹𝑘))
2422, 23fmptd 6425 . . . . . 6 ((𝜑𝑗𝐴) → (𝑘𝐶 ↦ (𝑗𝐹𝑘)):𝐶𝐵)
25 tsmsxp.1 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
26 df-ima 5156 . . . . . . . 8 ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿)
279, 8tgptopon 21933 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵))
2813, 27syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝐵))
29 tsmsxp.l . . . . . . . . . . . 12 (𝜑𝐿𝐽)
30 toponss 20779 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐿𝐽) → 𝐿𝐵)
3128, 29, 30syl2anc 694 . . . . . . . . . . 11 (𝜑𝐿𝐵)
3231adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐿𝐵)
3332resmptd 5487 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3433rneqd 5385 . . . . . . . 8 ((𝜑𝑗𝐴) → ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3526, 34syl5eq 2697 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
36 tsmsxp.h . . . . . . . . . . . . 13 (𝜑𝐻:𝐴𝐵)
3736ffvelrnda 6399 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ 𝐵)
38 tsmsxp.p . . . . . . . . . . . . 13 + = (+g𝐺)
39 eqid 2651 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
40 tsmsxp.m . . . . . . . . . . . . 13 = (-g𝐺)
418, 38, 39, 40grpsubval 17512 . . . . . . . . . . . 12 (((𝐻𝑗) ∈ 𝐵𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4237, 41sylan 487 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4342mpteq2dva 4777 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
44 tgpgrp 21929 . . . . . . . . . . . . . 14 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4513, 44syl 17 . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Grp)
4645adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → 𝐺 ∈ Grp)
478, 39grpinvcl 17514 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
4846, 47sylan 487 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
498, 39grpinvf 17513 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (invg𝐺):𝐵𝐵)
5046, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (invg𝐺):𝐵𝐵)
5150feqmptd 6288 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (invg𝐺) = (𝑔𝐵 ↦ ((invg𝐺)‘𝑔)))
52 eqidd 2652 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)))
53 oveq2 6698 . . . . . . . . . . 11 (𝑦 = ((invg𝐺)‘𝑔) → ((𝐻𝑗) + 𝑦) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
5448, 51, 52, 53fmptco 6436 . . . . . . . . . 10 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
5543, 54eqtr4d 2688 . . . . . . . . 9 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)))
5613adantr 480 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝐺 ∈ TopGrp)
579, 39grpinvhmeo 21937 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽Homeo𝐽))
5856, 57syl 17 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (invg𝐺) ∈ (𝐽Homeo𝐽))
59 eqid 2651 . . . . . . . . . . . 12 (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦))
6059, 8, 38, 9tgplacthmeo 21954 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ (𝐻𝑗) ∈ 𝐵) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
6156, 37, 60syl2anc 694 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
62 hmeoco 21623 . . . . . . . . . 10 (((invg𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽)) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6358, 61, 62syl2anc 694 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6455, 63eqeltrd 2730 . . . . . . . 8 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽))
6529adantr 480 . . . . . . . 8 ((𝜑𝑗𝐴) → 𝐿𝐽)
66 hmeoima 21616 . . . . . . . 8 (((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽) ∧ 𝐿𝐽) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6764, 65, 66syl2anc 694 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6835, 67eqeltrrd 2731 . . . . . 6 ((𝜑𝑗𝐴) → ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ∈ 𝐽)
69 tsmsxp.z . . . . . . . . 9 0 = (0g𝐺)
708, 69, 40grpsubid1 17547 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐻𝑗) ∈ 𝐵) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
7146, 37, 70syl2anc 694 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
72 tsmsxp.3 . . . . . . . . 9 (𝜑0𝐿)
7372adantr 480 . . . . . . . 8 ((𝜑𝑗𝐴) → 0𝐿)
74 ovex 6718 . . . . . . . 8 ((𝐻𝑗) 0 ) ∈ V
75 eqid 2651 . . . . . . . . 9 (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))
76 oveq2 6698 . . . . . . . . 9 (𝑔 = 0 → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) 0 ))
7775, 76elrnmpt1s 5405 . . . . . . . 8 (( 0𝐿 ∧ ((𝐻𝑗) 0 ) ∈ V) → ((𝐻𝑗) 0 ) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
7873, 74, 77sylancl 695 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐻𝑗) 0 ) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
7971, 78eqeltrrd 2731 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
808, 9, 10, 12, 16, 18, 24, 25, 68, 79tsmsi 21984 . . . . 5 ((𝜑𝑗𝐴) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
817, 80syldan 486 . . . 4 ((𝜑𝑗𝐾) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
8281ralrimiva 2995 . . 3 (𝜑 → ∀𝑗𝐾𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
83 sseq1 3659 . . . . . 6 (𝑦 = (𝑓𝑗) → (𝑦𝑧 ↔ (𝑓𝑗) ⊆ 𝑧))
8483imbi1d 330 . . . . 5 (𝑦 = (𝑓𝑗) → ((𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) ↔ ((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))))
8584ralbidv 3015 . . . 4 (𝑦 = (𝑓𝑗) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) ↔ ∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))))
8685ac6sfi 8245 . . 3 ((𝐾 ∈ Fin ∧ ∀𝑗𝐾𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))) → ∃𝑓(𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))))
874, 82, 86syl2anc 694 . 2 (𝜑 → ∃𝑓(𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))))
88 frn 6091 . . . . . . . . 9 (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
8988adantl 481 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
90 inss1 3866 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
9189, 90syl6ss 3648 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ 𝒫 𝐶)
92 sspwuni 4643 . . . . . . 7 (ran 𝑓 ⊆ 𝒫 𝐶 ran 𝑓𝐶)
9391, 92sylib 208 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓𝐶)
94 tsmsxp.d . . . . . . . . 9 (𝜑𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
95 elfpw 8309 . . . . . . . . . 10 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝐷 ⊆ (𝐴 × 𝐶) ∧ 𝐷 ∈ Fin))
9695simplbi 475 . . . . . . . . 9 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ⊆ (𝐴 × 𝐶))
97 rnss 5386 . . . . . . . . 9 (𝐷 ⊆ (𝐴 × 𝐶) → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
9894, 96, 973syl 18 . . . . . . . 8 (𝜑 → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
99 rnxpss 5601 . . . . . . . 8 ran (𝐴 × 𝐶) ⊆ 𝐶
10098, 99syl6ss 3648 . . . . . . 7 (𝜑 → ran 𝐷𝐶)
101100adantr 480 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷𝐶)
10293, 101unssd 3822 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
1034adantr 480 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐾 ∈ Fin)
104 ffn 6083 . . . . . . . . . 10 (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → 𝑓 Fn 𝐾)
105104adantl 481 . . . . . . . . 9 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓 Fn 𝐾)
106 dffn4 6159 . . . . . . . . 9 (𝑓 Fn 𝐾𝑓:𝐾onto→ran 𝑓)
107105, 106sylib 208 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓:𝐾onto→ran 𝑓)
108 fofi 8293 . . . . . . . 8 ((𝐾 ∈ Fin ∧ 𝑓:𝐾onto→ran 𝑓) → ran 𝑓 ∈ Fin)
109103, 107, 108syl2anc 694 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
110 inss2 3867 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ Fin
11189, 110syl6ss 3648 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ Fin)
112 unifi 8296 . . . . . . 7 ((ran 𝑓 ∈ Fin ∧ ran 𝑓 ⊆ Fin) → ran 𝑓 ∈ Fin)
113109, 111, 112syl2anc 694 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
11495simprbi 479 . . . . . . . 8 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ∈ Fin)
115 rnfi 8290 . . . . . . . 8 (𝐷 ∈ Fin → ran 𝐷 ∈ Fin)
11694, 114, 1153syl 18 . . . . . . 7 (𝜑 → ran 𝐷 ∈ Fin)
117116adantr 480 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷 ∈ Fin)
118 unfi 8268 . . . . . 6 (( ran 𝑓 ∈ Fin ∧ ran 𝐷 ∈ Fin) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
119113, 117, 118syl2anc 694 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
120 elfpw 8309 . . . . 5 (( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) ↔ (( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶 ∧ ( ran 𝑓 ∪ ran 𝐷) ∈ Fin))
121102, 119, 120sylanbrc 699 . . . 4 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
122121adantrr 753 . . 3 ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
123 ssun2 3810 . . . 4 ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷)
124123a1i 11 . . 3 ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))) → ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷))
125121adantlr 751 . . . . . . . . 9 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
126 fvssunirn 6255 . . . . . . . . . . . . . 14 (𝑓𝑗) ⊆ ran 𝑓
127 ssun1 3809 . . . . . . . . . . . . . 14 ran 𝑓 ⊆ ( ran 𝑓 ∪ ran 𝐷)
128126, 127sstri 3645 . . . . . . . . . . . . 13 (𝑓𝑗) ⊆ ( ran 𝑓 ∪ ran 𝐷)
129 id 22 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → 𝑧 = ( ran 𝑓 ∪ ran 𝐷))
130128, 129syl5sseqr 3687 . . . . . . . . . . . 12 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (𝑓𝑗) ⊆ 𝑧)
131 pm5.5 350 . . . . . . . . . . . 12 ((𝑓𝑗) ⊆ 𝑧 → (((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) ↔ (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
132130, 131syl 17 . . . . . . . . . . 11 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) ↔ (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
133 reseq2 5423 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧) = ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)))
134133oveq2d 6706 . . . . . . . . . . . 12 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) = (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))))
135134eleq1d 2715 . . . . . . . . . . 11 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → ((𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ↔ (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
136132, 135bitrd 268 . . . . . . . . . 10 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) ↔ (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
137136rspcv 3336 . . . . . . . . 9 (( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
138125, 137syl 17 . . . . . . . 8 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
13911ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
140 cmnmnd 18254 . . . . . . . . . . . . 13 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
141139, 140syl 17 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ Mnd)
142 simplr 807 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐾)
143119adantlr 751 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
144102adantlr 751 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
145144sselda 3636 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → 𝑘𝐶)
14619adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝐾) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
147146, 7jca 553 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝐾) → (𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴))
148203expa 1284 . . . . . . . . . . . . . . . . 17 (((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
149147, 148sylan 487 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
150149adantlr 751 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
151145, 150syldan 486 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ 𝐵)
152 eqid 2651 . . . . . . . . . . . . . 14 (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))
153151, 152fmptd 6425 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)):( ran 𝑓 ∪ ran 𝐷)⟶𝐵)
154 ovexd 6720 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ V)
155 fvex 6239 . . . . . . . . . . . . . . . 16 (0g𝐺) ∈ V
15669, 155eqeltri 2726 . . . . . . . . . . . . . . 15 0 ∈ V
157156a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 0 ∈ V)
158152, 143, 154, 157fsuppmptdm 8327 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) finSupp 0 )
1598, 69, 139, 143, 153, 158gsumcl 18362 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
160 velsn 4226 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑗} ↔ 𝑦 = 𝑗)
161 ovres 6842 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ {𝑗} ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
162160, 161sylanbr 489 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
163 oveq1 6697 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
164163adantr 480 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
165162, 164eqtrd 2685 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑗𝐹𝑘))
166165mpteq2dva 4777 . . . . . . . . . . . . . 14 (𝑦 = 𝑗 → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
167166oveq2d 6706 . . . . . . . . . . . . 13 (𝑦 = 𝑗 → (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘))) = (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
1688, 167gsumsn 18400 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑗𝐾 ∧ (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) → (𝐺 Σg (𝑦 ∈ {𝑗} ↦ (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘))))) = (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
169141, 142, 159, 168syl3anc 1366 . . . . . . . . . . 11 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝑦 ∈ {𝑗} ↦ (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘))))) = (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
170 snfi 8079 . . . . . . . . . . . . 13 {𝑗} ∈ Fin
171170a1i 11 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ∈ Fin)
17219ad2antrr 762 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
1737adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐴)
174173snssd 4372 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ⊆ 𝐴)
175 xpss12 5158 . . . . . . . . . . . . . 14 (({𝑗} ⊆ 𝐴 ∧ ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
176174, 144, 175syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
177172, 176fssresd 6109 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))):({𝑗} × ( ran 𝑓 ∪ ran 𝐷))⟶𝐵)
178 xpfi 8272 . . . . . . . . . . . . . 14 (({𝑗} ∈ Fin ∧ ( ran 𝑓 ∪ ran 𝐷) ∈ Fin) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
179170, 143, 178sylancr 696 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
180177, 179, 157fdmfifsupp 8326 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) finSupp 0 )
1818, 69, 139, 171, 143, 177, 180gsumxp 18421 . . . . . . . . . . 11 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = (𝐺 Σg (𝑦 ∈ {𝑗} ↦ (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘))))))
182144resmptd 5487 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
183182oveq2d 6706 . . . . . . . . . . 11 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
184169, 181, 1833eqtr4rd 2696 . . . . . . . . . 10 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))))
185184eleq1d 2715 . . . . . . . . 9 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ↔ (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
186 ovex 6718 . . . . . . . . . . 11 ((𝐻𝑗) 𝑔) ∈ V
18775, 186elrnmpti 5408 . . . . . . . . . 10 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ↔ ∃𝑔𝐿 (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔))
188 isabl 18243 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
18945, 11, 188sylanbrc 699 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Abel)
190189ad3antrrr 766 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝐺 ∈ Abel)
1917, 37syldan 486 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝐾) → (𝐻𝑗) ∈ 𝐵)
192191ad2antrr 762 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → (𝐻𝑗) ∈ 𝐵)
19331ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐿𝐵)
194193sselda 3636 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐵)
1958, 40, 190, 192, 194ablnncan 18272 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) = 𝑔)
196 simpr 476 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐿)
197195, 196eqeltrd 2730 . . . . . . . . . . . 12 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) ∈ 𝐿)
198 oveq2 6698 . . . . . . . . . . . . 13 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑗) ((𝐻𝑗) 𝑔)))
199198eleq1d 2715 . . . . . . . . . . . 12 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔) → (((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ((𝐻𝑗) ((𝐻𝑗) 𝑔)) ∈ 𝐿))
200197, 199syl5ibrcom 237 . . . . . . . . . . 11 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
201200rexlimdva 3060 . . . . . . . . . 10 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∃𝑔𝐿 (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
202187, 201syl5bi 232 . . . . . . . . 9 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
203185, 202sylbid 230 . . . . . . . 8 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
204138, 203syld 47 . . . . . . 7 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
205204an32s 863 . . . . . 6 (((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑗𝐾) → (∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
206205ralimdva 2991 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) → ∀𝑗𝐾 ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
207206impr 648 . . . 4 ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))) → ∀𝑗𝐾 ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
208 fveq2 6229 . . . . . . 7 (𝑗 = 𝑥 → (𝐻𝑗) = (𝐻𝑥))
209 sneq 4220 . . . . . . . . . 10 (𝑗 = 𝑥 → {𝑗} = {𝑥})
210209xpeq1d 5172 . . . . . . . . 9 (𝑗 = 𝑥 → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
211210reseq2d 5428 . . . . . . . 8 (𝑗 = 𝑥 → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
212211oveq2d 6706 . . . . . . 7 (𝑗 = 𝑥 → (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
213208, 212oveq12d 6708 . . . . . 6 (𝑗 = 𝑥 → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
214213eleq1d 2715 . . . . 5 (𝑗 = 𝑥 → (((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
215214cbvralv 3201 . . . 4 (∀𝑗𝐾 ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
216207, 215sylib 208 . . 3 ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))) → ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
217 sseq2 3660 . . . . 5 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (ran 𝐷𝑛 ↔ ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷)))
218 xpeq2 5163 . . . . . . . . . 10 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ({𝑥} × 𝑛) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
219218reseq2d 5428 . . . . . . . . 9 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐹 ↾ ({𝑥} × 𝑛)) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
220219oveq2d 6706 . . . . . . . 8 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
221220oveq2d 6706 . . . . . . 7 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
222221eleq1d 2715 . . . . . 6 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
223222ralbidv 3015 . . . . 5 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿 ↔ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
224217, 223anbi12d 747 . . . 4 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ((ran 𝐷𝑛 ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿) ↔ (ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷) ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)))
225224rspcev 3340 . . 3 ((( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) ∧ (ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷) ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)) → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷𝑛 ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
226122, 124, 216, 225syl12anc 1364 . 2 ((𝜑 ∧ (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))) → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷𝑛 ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
22787, 226exlimddv 1903 1 (𝜑 → ∃𝑛 ∈ (𝒫 𝐶 ∩ Fin)(ran 𝐷𝑛 ∧ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cun 3605  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  cres 5145  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  Fincfn 7997  Basecbs 15904  +gcplusg 15988  TopOpenctopn 16129  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  Grpcgrp 17469  invgcminusg 17470  -gcsg 17471  CMndccmn 18239  Abelcabl 18240  TopOnctopon 20763  TopSpctps 20784  Homeochmeo 21604  TopGrpctgp 21922   tsums ctsu 21976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-topgen 16151  df-mre 16293  df-mrc 16294  df-acs 16296  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-abl 18242  df-fbas 19791  df-fg 19792  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-ntr 20872  df-nei 20950  df-cn 21079  df-cnp 21080  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-tmd 21923  df-tgp 21924  df-tsms 21977
This theorem is referenced by:  tsmsxp  22005
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