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Theorem tsmsxplem1 21866
Description: Lemma for tsmsxp 21868. (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 8212 . . . . 5 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝐾𝐴𝐾 ∈ Fin))
32simprbi 480 . . . 4 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) → 𝐾 ∈ Fin)
41, 3syl 17 . . 3 (𝜑𝐾 ∈ Fin)
52simplbi 476 . . . . . . 7 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) → 𝐾𝐴)
61, 5syl 17 . . . . . 6 (𝜑𝐾𝐴)
76sselda 3583 . . . . 5 ((𝜑𝑗𝐾) → 𝑗𝐴)
8 tsmsxp.b . . . . . 6 𝐵 = (Base‘𝐺)
9 tsmsxp.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
10 eqid 2621 . . . . . 6 (𝒫 𝐶 ∩ Fin) = (𝒫 𝐶 ∩ Fin)
11 tsmsxp.g . . . . . . 7 (𝜑𝐺 ∈ CMnd)
1211adantr 481 . . . . . 6 ((𝜑𝑗𝐴) → 𝐺 ∈ CMnd)
13 tsmsxp.2 . . . . . . . 8 (𝜑𝐺 ∈ TopGrp)
14 tgptps 21794 . . . . . . . 8 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
1513, 14syl 17 . . . . . . 7 (𝜑𝐺 ∈ TopSp)
1615adantr 481 . . . . . 6 ((𝜑𝑗𝐴) → 𝐺 ∈ TopSp)
17 tsmsxp.c . . . . . . 7 (𝜑𝐶𝑊)
1817adantr 481 . . . . . 6 ((𝜑𝑗𝐴) → 𝐶𝑊)
19 tsmsxp.f . . . . . . . . 9 (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)
20 fovrn 6757 . . . . . . . . 9 ((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
2119, 20syl3an1 1356 . . . . . . . 8 ((𝜑𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
22213expa 1262 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
23 eqid 2621 . . . . . . 7 (𝑘𝐶 ↦ (𝑗𝐹𝑘)) = (𝑘𝐶 ↦ (𝑗𝐹𝑘))
2422, 23fmptd 6340 . . . . . 6 ((𝜑𝑗𝐴) → (𝑘𝐶 ↦ (𝑗𝐹𝑘)):𝐶𝐵)
25 tsmsxp.1 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
26 df-ima 5087 . . . . . . . 8 ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿)
279, 8tgptopon 21796 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵))
2813, 27syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝐵))
29 tsmsxp.l . . . . . . . . . . . 12 (𝜑𝐿𝐽)
30 toponss 20644 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐿𝐽) → 𝐿𝐵)
3128, 29, 30syl2anc 692 . . . . . . . . . . 11 (𝜑𝐿𝐵)
3231adantr 481 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐿𝐵)
3332resmptd 5411 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3433rneqd 5313 . . . . . . . 8 ((𝜑𝑗𝐴) → ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3526, 34syl5eq 2667 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
36 tsmsxp.h . . . . . . . . . . . . 13 (𝜑𝐻:𝐴𝐵)
3736ffvelrnda 6315 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ 𝐵)
38 tsmsxp.p . . . . . . . . . . . . 13 + = (+g𝐺)
39 eqid 2621 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
40 tsmsxp.m . . . . . . . . . . . . 13 = (-g𝐺)
418, 38, 39, 40grpsubval 17386 . . . . . . . . . . . 12 (((𝐻𝑗) ∈ 𝐵𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4237, 41sylan 488 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4342mpteq2dva 4704 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
44 tgpgrp 21792 . . . . . . . . . . . . . 14 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4513, 44syl 17 . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Grp)
4645adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → 𝐺 ∈ Grp)
478, 39grpinvcl 17388 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
4846, 47sylan 488 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
498, 39grpinvf 17387 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (invg𝐺):𝐵𝐵)
5046, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (invg𝐺):𝐵𝐵)
5150feqmptd 6206 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (invg𝐺) = (𝑔𝐵 ↦ ((invg𝐺)‘𝑔)))
52 eqidd 2622 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)))
53 oveq2 6612 . . . . . . . . . . 11 (𝑦 = ((invg𝐺)‘𝑔) → ((𝐻𝑗) + 𝑦) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
5448, 51, 52, 53fmptco 6351 . . . . . . . . . 10 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
5543, 54eqtr4d 2658 . . . . . . . . 9 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)))
5613adantr 481 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝐺 ∈ TopGrp)
579, 39grpinvhmeo 21800 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽Homeo𝐽))
5856, 57syl 17 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (invg𝐺) ∈ (𝐽Homeo𝐽))
59 eqid 2621 . . . . . . . . . . . 12 (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦))
6059, 8, 38, 9tgplacthmeo 21817 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ (𝐻𝑗) ∈ 𝐵) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
6156, 37, 60syl2anc 692 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
62 hmeoco 21485 . . . . . . . . . 10 (((invg𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽)) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6358, 61, 62syl2anc 692 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6455, 63eqeltrd 2698 . . . . . . . 8 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽))
6529adantr 481 . . . . . . . 8 ((𝜑𝑗𝐴) → 𝐿𝐽)
66 hmeoima 21478 . . . . . . . 8 (((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽) ∧ 𝐿𝐽) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6764, 65, 66syl2anc 692 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6835, 67eqeltrrd 2699 . . . . . 6 ((𝜑𝑗𝐴) → ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ∈ 𝐽)
69 tsmsxp.z . . . . . . . . 9 0 = (0g𝐺)
708, 69, 40grpsubid1 17421 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐻𝑗) ∈ 𝐵) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
7146, 37, 70syl2anc 692 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
72 tsmsxp.3 . . . . . . . . 9 (𝜑0𝐿)
7372adantr 481 . . . . . . . 8 ((𝜑𝑗𝐴) → 0𝐿)
74 ovex 6632 . . . . . . . 8 ((𝐻𝑗) 0 ) ∈ V
75 eqid 2621 . . . . . . . . 9 (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))
76 oveq2 6612 . . . . . . . . 9 (𝑔 = 0 → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) 0 ))
7775, 76elrnmpt1s 5333 . . . . . . . 8 (( 0𝐿 ∧ ((𝐻𝑗) 0 ) ∈ V) → ((𝐻𝑗) 0 ) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
7873, 74, 77sylancl 693 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐻𝑗) 0 ) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
7971, 78eqeltrrd 2699 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
808, 9, 10, 12, 16, 18, 24, 25, 68, 79tsmsi 21847 . . . . 5 ((𝜑𝑗𝐴) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
817, 80syldan 487 . . . 4 ((𝜑𝑗𝐾) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
8281ralrimiva 2960 . . 3 (𝜑 → ∀𝑗𝐾𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
83 sseq1 3605 . . . . . 6 (𝑦 = (𝑓𝑗) → (𝑦𝑧 ↔ (𝑓𝑗) ⊆ 𝑧))
8483imbi1d 331 . . . . 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 2980 . . . 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 8148 . . 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 692 . 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 6010 . . . . . . . . 9 (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
8988adantl 482 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
90 inss1 3811 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
9189, 90syl6ss 3595 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ 𝒫 𝐶)
92 sspwuni 4577 . . . . . . 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 8212 . . . . . . . . . 10 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝐷 ⊆ (𝐴 × 𝐶) ∧ 𝐷 ∈ Fin))
9695simplbi 476 . . . . . . . . 9 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ⊆ (𝐴 × 𝐶))
97 rnss 5314 . . . . . . . . 9 (𝐷 ⊆ (𝐴 × 𝐶) → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
9894, 96, 973syl 18 . . . . . . . 8 (𝜑 → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
99 rnxpss 5525 . . . . . . . 8 ran (𝐴 × 𝐶) ⊆ 𝐶
10098, 99syl6ss 3595 . . . . . . 7 (𝜑 → ran 𝐷𝐶)
101100adantr 481 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷𝐶)
10293, 101unssd 3767 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
1034adantr 481 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐾 ∈ Fin)
104 ffn 6002 . . . . . . . . . 10 (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → 𝑓 Fn 𝐾)
105104adantl 482 . . . . . . . . 9 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓 Fn 𝐾)
106 dffn4 6078 . . . . . . . . 9 (𝑓 Fn 𝐾𝑓:𝐾onto→ran 𝑓)
107105, 106sylib 208 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓:𝐾onto→ran 𝑓)
108 fofi 8196 . . . . . . . 8 ((𝐾 ∈ Fin ∧ 𝑓:𝐾onto→ran 𝑓) → ran 𝑓 ∈ Fin)
109103, 107, 108syl2anc 692 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
110 inss2 3812 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ Fin
11189, 110syl6ss 3595 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ Fin)
112 unifi 8199 . . . . . . 7 ((ran 𝑓 ∈ Fin ∧ ran 𝑓 ⊆ Fin) → ran 𝑓 ∈ Fin)
113109, 111, 112syl2anc 692 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
11495simprbi 480 . . . . . . . 8 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ∈ Fin)
115 rnfi 8193 . . . . . . . 8 (𝐷 ∈ Fin → ran 𝐷 ∈ Fin)
11694, 114, 1153syl 18 . . . . . . 7 (𝜑 → ran 𝐷 ∈ Fin)
117116adantr 481 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷 ∈ Fin)
118 unfi 8171 . . . . . 6 (( ran 𝑓 ∈ Fin ∧ ran 𝐷 ∈ Fin) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
119113, 117, 118syl2anc 692 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
120 elfpw 8212 . . . . 5 (( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) ↔ (( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶 ∧ ( ran 𝑓 ∪ ran 𝐷) ∈ Fin))
121102, 119, 120sylanbrc 697 . . . 4 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
122121adantrr 752 . . 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 3755 . . . 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 750 . . . . . . . . 9 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
126 fvssunirn 6174 . . . . . . . . . . . . . 14 (𝑓𝑗) ⊆ ran 𝑓
127 ssun1 3754 . . . . . . . . . . . . . 14 ran 𝑓 ⊆ ( ran 𝑓 ∪ ran 𝐷)
128126, 127sstri 3592 . . . . . . . . . . . . 13 (𝑓𝑗) ⊆ ( ran 𝑓 ∪ ran 𝐷)
129 id 22 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → 𝑧 = ( ran 𝑓 ∪ ran 𝐷))
130128, 129syl5sseqr 3633 . . . . . . . . . . . 12 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (𝑓𝑗) ⊆ 𝑧)
131 pm5.5 351 . . . . . . . . . . . 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 5351 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧) = ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)))
134133oveq2d 6620 . . . . . . . . . . . 12 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) = (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))))
135134eleq1d 2683 . . . . . . . . . . 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 3291 . . . . . . . . 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 761 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
140 cmnmnd 18129 . . . . . . . . . . . . 13 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
141139, 140syl 17 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ Mnd)
142 simplr 791 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐾)
143119adantlr 750 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
144102adantlr 750 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
145144sselda 3583 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → 𝑘𝐶)
14619adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝐾) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
147146, 7jca 554 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝐾) → (𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴))
148203expa 1262 . . . . . . . . . . . . . . . . 17 (((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
149147, 148sylan 488 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
150149adantlr 750 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
151145, 150syldan 487 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ 𝐵)
152 eqid 2621 . . . . . . . . . . . . . 14 (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))
153151, 152fmptd 6340 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)):( ran 𝑓 ∪ ran 𝐷)⟶𝐵)
154 ovex 6632 . . . . . . . . . . . . . . 15 (𝑗𝐹𝑘) ∈ V
155154a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ V)
156 fvex 6158 . . . . . . . . . . . . . . . 16 (0g𝐺) ∈ V
15769, 156eqeltri 2694 . . . . . . . . . . . . . . 15 0 ∈ V
158157a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 0 ∈ V)
159152, 143, 155, 158fsuppmptdm 8230 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) finSupp 0 )
1608, 69, 139, 143, 153, 159gsumcl 18237 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
161 velsn 4164 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑗} ↔ 𝑦 = 𝑗)
162 ovres 6753 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ {𝑗} ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
163161, 162sylanbr 490 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
164 oveq1 6611 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
165164adantr 481 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
166163, 165eqtrd 2655 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑗𝐹𝑘))
167166mpteq2dva 4704 . . . . . . . . . . . . . 14 (𝑦 = 𝑗 → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
168167oveq2d 6620 . . . . . . . . . . . . 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))))
1698, 168gsumsn 18275 . . . . . . . . . . . 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))))
170141, 142, 160, 169syl3anc 1323 . . . . . . . . . . 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))))
171 snfi 7982 . . . . . . . . . . . . 13 {𝑗} ∈ Fin
172171a1i 11 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ∈ Fin)
17319ad2antrr 761 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
1747adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐴)
175174snssd 4309 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ⊆ 𝐴)
176 xpss12 5186 . . . . . . . . . . . . . 14 (({𝑗} ⊆ 𝐴 ∧ ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
177175, 144, 176syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
178173, 177fssresd 6028 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))):({𝑗} × ( ran 𝑓 ∪ ran 𝐷))⟶𝐵)
179 xpfi 8175 . . . . . . . . . . . . . 14 (({𝑗} ∈ Fin ∧ ( ran 𝑓 ∪ ran 𝐷) ∈ Fin) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
180171, 143, 179sylancr 694 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
181178, 180, 158fdmfifsupp 8229 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) finSupp 0 )
1828, 69, 139, 172, 143, 178, 181gsumxp 18296 . . . . . . . . . . 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))))))
183144resmptd 5411 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
184183oveq2d 6620 . . . . . . . . . . 11 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
185170, 182, 1843eqtr4rd 2666 . . . . . . . . . 10 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))))
186185eleq1d 2683 . . . . . . . . 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))))
187 ovex 6632 . . . . . . . . . . 11 ((𝐻𝑗) 𝑔) ∈ V
18875, 187elrnmpti 5336 . . . . . . . . . 10 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ↔ ∃𝑔𝐿 (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔))
189 isabl 18118 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
19045, 11, 189sylanbrc 697 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Abel)
191190ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝐺 ∈ Abel)
1927, 37syldan 487 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝐾) → (𝐻𝑗) ∈ 𝐵)
193192ad2antrr 761 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → (𝐻𝑗) ∈ 𝐵)
19431ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐿𝐵)
195194sselda 3583 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐵)
1968, 40, 191, 193, 195ablnncan 18147 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) = 𝑔)
197 simpr 477 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐿)
198196, 197eqeltrd 2698 . . . . . . . . . . . 12 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) ∈ 𝐿)
199 oveq2 6612 . . . . . . . . . . . . 13 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑗) ((𝐻𝑗) 𝑔)))
200199eleq1d 2683 . . . . . . . . . . . 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))
201198, 200syl5ibrcom 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))
202201rexlimdva 3024 . . . . . . . . . 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))
203188, 202syl5bi 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))
204186, 203sylbid 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))
205138, 204syld 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))
206205an32s 845 . . . . . 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))
207206ralimdva 2956 . . . . 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))
208207impr 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)
209 fveq2 6148 . . . . . . 7 (𝑗 = 𝑥 → (𝐻𝑗) = (𝐻𝑥))
210 sneq 4158 . . . . . . . . . 10 (𝑗 = 𝑥 → {𝑗} = {𝑥})
211210xpeq1d 5098 . . . . . . . . 9 (𝑗 = 𝑥 → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
212211reseq2d 5356 . . . . . . . 8 (𝑗 = 𝑥 → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
213212oveq2d 6620 . . . . . . 7 (𝑗 = 𝑥 → (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
214209, 213oveq12d 6622 . . . . . 6 (𝑗 = 𝑥 → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
215214eleq1d 2683 . . . . 5 (𝑗 = 𝑥 → (((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
216215cbvralv 3159 . . . 4 (∀𝑗𝐾 ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
217208, 216sylib 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)
218 sseq2 3606 . . . . 5 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (ran 𝐷𝑛 ↔ ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷)))
219 xpeq2 5089 . . . . . . . . . 10 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ({𝑥} × 𝑛) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
220219reseq2d 5356 . . . . . . . . 9 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐹 ↾ ({𝑥} × 𝑛)) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
221220oveq2d 6620 . . . . . . . 8 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
222221oveq2d 6620 . . . . . . 7 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
223222eleq1d 2683 . . . . . 6 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
224223ralbidv 2980 . . . . 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))
225218, 224anbi12d 746 . . . 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)))
226225rspcev 3295 . . 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))
227122, 124, 217, 226syl12anc 1321 . 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))
22887, 227exlimddv 1860 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 384   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cun 3553  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402  cmpt 4673   × cxp 5072  dom cdm 5074  ran crn 5075  cres 5076  cima 5077  ccom 5078   Fn wfn 5842  wf 5843  ontowfo 5845  cfv 5847  (class class class)co 6604  Fincfn 7899  Basecbs 15781  +gcplusg 15862  TopOpenctopn 16003  0gc0g 16021   Σg cgsu 16022  Mndcmnd 17215  Grpcgrp 17343  invgcminusg 17344  -gcsg 17345  CMndccmn 18114  Abelcabl 18115  TopOnctopon 20618  TopSpctps 20619  Homeochmeo 21466  TopGrpctgp 21785   tsums ctsu 21839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-0g 16023  df-gsum 16024  df-topgen 16025  df-mre 16167  df-mrc 16168  df-acs 16170  df-plusf 17162  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-abl 18117  df-fbas 19662  df-fg 19663  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-ntr 20734  df-nei 20812  df-cn 20941  df-cnp 20942  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-tmd 21786  df-tgp 21787  df-tsms 21840
This theorem is referenced by:  tsmsxp  21868
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