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Theorem tsmsxplem1 21766
Description: Lemma for tsmsxp 21768. (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 8151 . . . . 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 3568 . . . . 5 ((𝜑𝑗𝐾) → 𝑗𝐴)
8 tsmsxp.b . . . . . 6 𝐵 = (Base‘𝐺)
9 tsmsxp.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
10 eqid 2610 . . . . . 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 21694 . . . . . . . 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 6702 . . . . . . . . 9 ((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
2119, 20syl3an1 1351 . . . . . . . 8 ((𝜑𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
22213expa 1257 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
23 eqid 2610 . . . . . . 7 (𝑘𝐶 ↦ (𝑗𝐹𝑘)) = (𝑘𝐶 ↦ (𝑗𝐹𝑘))
2422, 23fmptd 6292 . . . . . 6 ((𝜑𝑗𝐴) → (𝑘𝐶 ↦ (𝑗𝐹𝑘)):𝐶𝐵)
25 tsmsxp.1 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
26 df-ima 5051 . . . . . . . 8 ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿)
279, 8tgptopon 21696 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵))
2813, 27syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝐵))
29 tsmsxp.l . . . . . . . . . . . 12 (𝜑𝐿𝐽)
30 toponss 20544 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐿𝐽) → 𝐿𝐵)
3128, 29, 30syl2anc 691 . . . . . . . . . . 11 (𝜑𝐿𝐵)
3231adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐿𝐵)
3332resmptd 5371 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3433rneqd 5274 . . . . . . . 8 ((𝜑𝑗𝐴) → ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3526, 34syl5eq 2656 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
36 tsmsxp.h . . . . . . . . . . . . 13 (𝜑𝐻:𝐴𝐵)
3736ffvelrnda 6267 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ 𝐵)
38 tsmsxp.p . . . . . . . . . . . . 13 + = (+g𝐺)
39 eqid 2610 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
40 tsmsxp.m . . . . . . . . . . . . 13 = (-g𝐺)
418, 38, 39, 40grpsubval 17288 . . . . . . . . . . . 12 (((𝐻𝑗) ∈ 𝐵𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4237, 41sylan 487 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4342mpteq2dva 4672 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
44 tgpgrp 21692 . . . . . . . . . . . . . 14 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4513, 44syl 17 . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Grp)
4645adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → 𝐺 ∈ Grp)
478, 39grpinvcl 17290 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
4846, 47sylan 487 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
498, 39grpinvf 17289 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (invg𝐺):𝐵𝐵)
5046, 49syl 17 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (invg𝐺):𝐵𝐵)
5150feqmptd 6159 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (invg𝐺) = (𝑔𝐵 ↦ ((invg𝐺)‘𝑔)))
52 eqidd 2611 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)))
53 oveq2 6557 . . . . . . . . . . 11 (𝑦 = ((invg𝐺)‘𝑔) → ((𝐻𝑗) + 𝑦) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
5448, 51, 52, 53fmptco 6303 . . . . . . . . . 10 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
5543, 54eqtr4d 2647 . . . . . . . . 9 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)))
5613adantr 480 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝐺 ∈ TopGrp)
579, 39grpinvhmeo 21700 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽Homeo𝐽))
5856, 57syl 17 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (invg𝐺) ∈ (𝐽Homeo𝐽))
59 eqid 2610 . . . . . . . . . . . 12 (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦))
6059, 8, 38, 9tgplacthmeo 21717 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ (𝐻𝑗) ∈ 𝐵) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
6156, 37, 60syl2anc 691 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
62 hmeoco 21385 . . . . . . . . . 10 (((invg𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽)) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6358, 61, 62syl2anc 691 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6455, 63eqeltrd 2688 . . . . . . . 8 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽))
6529adantr 480 . . . . . . . 8 ((𝜑𝑗𝐴) → 𝐿𝐽)
66 hmeoima 21378 . . . . . . . 8 (((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽) ∧ 𝐿𝐽) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6764, 65, 66syl2anc 691 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6835, 67eqeltrrd 2689 . . . . . 6 ((𝜑𝑗𝐴) → ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ∈ 𝐽)
69 tsmsxp.z . . . . . . . . 9 0 = (0g𝐺)
708, 69, 40grpsubid1 17323 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐻𝑗) ∈ 𝐵) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
7146, 37, 70syl2anc 691 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
72 tsmsxp.3 . . . . . . . . 9 (𝜑0𝐿)
7372adantr 480 . . . . . . . 8 ((𝜑𝑗𝐴) → 0𝐿)
74 ovex 6577 . . . . . . . 8 ((𝐻𝑗) 0 ) ∈ V
75 eqid 2610 . . . . . . . . 9 (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))
76 oveq2 6557 . . . . . . . . 9 (𝑔 = 0 → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) 0 ))
7775, 76elrnmpt1s 5294 . . . . . . . 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 2689 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
808, 9, 10, 12, 16, 18, 24, 25, 68, 79tsmsi 21747 . . . . 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 2949 . . 3 (𝜑 → ∀𝑗𝐾𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
83 sseq1 3589 . . . . . 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 2969 . . . 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 8089 . . 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 691 . 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 5966 . . . . . . . . 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 3795 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
9189, 90syl6ss 3580 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ 𝒫 𝐶)
92 sspwuni 4547 . . . . . . 7 (ran 𝑓 ⊆ 𝒫 𝐶 ran 𝑓𝐶)
9391, 92sylib 207 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓𝐶)
94 tsmsxp.d . . . . . . . . 9 (𝜑𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
95 elfpw 8151 . . . . . . . . . 10 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝐷 ⊆ (𝐴 × 𝐶) ∧ 𝐷 ∈ Fin))
9695simplbi 475 . . . . . . . . 9 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ⊆ (𝐴 × 𝐶))
97 rnss 5275 . . . . . . . . 9 (𝐷 ⊆ (𝐴 × 𝐶) → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
9894, 96, 973syl 18 . . . . . . . 8 (𝜑 → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
99 rnxpss 5485 . . . . . . . 8 ran (𝐴 × 𝐶) ⊆ 𝐶
10098, 99syl6ss 3580 . . . . . . 7 (𝜑 → ran 𝐷𝐶)
101100adantr 480 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷𝐶)
10293, 101unssd 3751 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
1034adantr 480 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐾 ∈ Fin)
104 ffn 5958 . . . . . . . . . 10 (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → 𝑓 Fn 𝐾)
105104adantl 481 . . . . . . . . 9 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓 Fn 𝐾)
106 dffn4 6034 . . . . . . . . 9 (𝑓 Fn 𝐾𝑓:𝐾onto→ran 𝑓)
107105, 106sylib 207 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓:𝐾onto→ran 𝑓)
108 fofi 8135 . . . . . . . 8 ((𝐾 ∈ Fin ∧ 𝑓:𝐾onto→ran 𝑓) → ran 𝑓 ∈ Fin)
109103, 107, 108syl2anc 691 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
110 inss2 3796 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ Fin
11189, 110syl6ss 3580 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ Fin)
112 unifi 8138 . . . . . . 7 ((ran 𝑓 ∈ Fin ∧ ran 𝑓 ⊆ Fin) → ran 𝑓 ∈ Fin)
113109, 111, 112syl2anc 691 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
11495simprbi 479 . . . . . . . 8 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ∈ Fin)
115 rnfi 8132 . . . . . . . 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 8112 . . . . . 6 (( ran 𝑓 ∈ Fin ∧ ran 𝐷 ∈ Fin) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
119113, 117, 118syl2anc 691 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
120 elfpw 8151 . . . . 5 (( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) ↔ (( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶 ∧ ( ran 𝑓 ∪ ran 𝐷) ∈ Fin))
121102, 119, 120sylanbrc 695 . . . 4 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
122121adantrr 749 . . 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 3739 . . . 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 747 . . . . . . . . 9 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
126 fvssunirn 6127 . . . . . . . . . . . . . 14 (𝑓𝑗) ⊆ ran 𝑓
127 ssun1 3738 . . . . . . . . . . . . . 14 ran 𝑓 ⊆ ( ran 𝑓 ∪ ran 𝐷)
128126, 127sstri 3577 . . . . . . . . . . . . 13 (𝑓𝑗) ⊆ ( ran 𝑓 ∪ ran 𝐷)
129 id 22 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → 𝑧 = ( ran 𝑓 ∪ ran 𝐷))
130128, 129syl5sseqr 3617 . . . . . . . . . . . 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 5312 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧) = ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)))
134133oveq2d 6565 . . . . . . . . . . . 12 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) = (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))))
135134eleq1d 2672 . . . . . . . . . . 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 267 . . . . . . . . . 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 3278 . . . . . . . . 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 758 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
140 cmnmnd 18031 . . . . . . . . . . . . 13 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
141139, 140syl 17 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ Mnd)
142 simplr 788 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐾)
143119adantlr 747 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
144102adantlr 747 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
145144sselda 3568 . . . . . . . . . . . . . . 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 1257 . . . . . . . . . . . . . . . . 17 (((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
149147, 148sylan 487 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
150149adantlr 747 . . . . . . . . . . . . . . 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 2610 . . . . . . . . . . . . . 14 (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))
153151, 152fmptd 6292 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)):( ran 𝑓 ∪ ran 𝐷)⟶𝐵)
154 ovex 6577 . . . . . . . . . . . . . . 15 (𝑗𝐹𝑘) ∈ V
155154a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ V)
156 fvex 6113 . . . . . . . . . . . . . . . 16 (0g𝐺) ∈ V
15769, 156eqeltri 2684 . . . . . . . . . . . . . . 15 0 ∈ V
158157a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 0 ∈ V)
159152, 143, 155, 158fsuppmptdm 8169 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) finSupp 0 )
1608, 69, 139, 143, 153, 159gsumcl 18139 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
161 velsn 4141 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑗} ↔ 𝑦 = 𝑗)
162 ovres 6698 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ {𝑗} ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
163161, 162sylanbr 489 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
164 oveq1 6556 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
165164adantr 480 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
166163, 165eqtrd 2644 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑗𝐹𝑘))
167166mpteq2dva 4672 . . . . . . . . . . . . . 14 (𝑦 = 𝑗 → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
168167oveq2d 6565 . . . . . . . . . . . . 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 18177 . . . . . . . . . . . 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 1318 . . . . . . . . . . 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 7923 . . . . . . . . . . . . 13 {𝑗} ∈ Fin
172171a1i 11 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ∈ Fin)
17319ad2antrr 758 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
1747adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐴)
175174snssd 4281 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ⊆ 𝐴)
176 xpss12 5148 . . . . . . . . . . . . . 14 (({𝑗} ⊆ 𝐴 ∧ ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
177175, 144, 176syl2anc 691 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
178173, 177fssresd 5984 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))):({𝑗} × ( ran 𝑓 ∪ ran 𝐷))⟶𝐵)
179 xpfi 8116 . . . . . . . . . . . . . 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 8168 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) finSupp 0 )
1828, 69, 139, 172, 143, 178, 181gsumxp 18198 . . . . . . . . . . 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 5371 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
184183oveq2d 6565 . . . . . . . . . . 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 2655 . . . . . . . . . 10 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))))
186185eleq1d 2672 . . . . . . . . 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 6577 . . . . . . . . . . 11 ((𝐻𝑗) 𝑔) ∈ V
18875, 187elrnmpti 5297 . . . . . . . . . 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 18020 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
19045, 11, 189sylanbrc 695 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Abel)
191190ad3antrrr 762 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝐺 ∈ Abel)
1927, 37syldan 486 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝐾) → (𝐻𝑗) ∈ 𝐵)
193192ad2antrr 758 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → (𝐻𝑗) ∈ 𝐵)
19431ad2antrr 758 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐿𝐵)
195194sselda 3568 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐵)
1968, 40, 191, 193, 195ablnncan 18049 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) = 𝑔)
197 simpr 476 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐿)
198196, 197eqeltrd 2688 . . . . . . . . . . . 12 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) ∈ 𝐿)
199 oveq2 6557 . . . . . . . . . . . . 13 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑗) ((𝐻𝑗) 𝑔)))
200199eleq1d 2672 . . . . . . . . . . . 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 236 . . . . . . . . . . 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 3013 . . . . . . . . . 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 231 . . . . . . . . 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 229 . . . . . . . 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 46 . . . . . . 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 842 . . . . . 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 2945 . . . . 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 647 . . . 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 6103 . . . . . . 7 (𝑗 = 𝑥 → (𝐻𝑗) = (𝐻𝑥))
210 sneq 4135 . . . . . . . . . 10 (𝑗 = 𝑥 → {𝑗} = {𝑥})
211210xpeq1d 5062 . . . . . . . . 9 (𝑗 = 𝑥 → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
212211reseq2d 5317 . . . . . . . 8 (𝑗 = 𝑥 → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
213212oveq2d 6565 . . . . . . 7 (𝑗 = 𝑥 → (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
214209, 213oveq12d 6567 . . . . . 6 (𝑗 = 𝑥 → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
215214eleq1d 2672 . . . . 5 (𝑗 = 𝑥 → (((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
216215cbvralv 3147 . . . 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 207 . . 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 3590 . . . . 5 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (ran 𝐷𝑛 ↔ ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷)))
219 xpeq2 5053 . . . . . . . . . 10 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ({𝑥} × 𝑛) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
220219reseq2d 5317 . . . . . . . . 9 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐹 ↾ ({𝑥} × 𝑛)) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
221220oveq2d 6565 . . . . . . . 8 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
222221oveq2d 6565 . . . . . . 7 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
223222eleq1d 2672 . . . . . 6 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
224223ralbidv 2969 . . . . 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 743 . . . 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 3282 . . 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 1316 . 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 1850 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 195  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  cun 3538  cin 3539  wss 3540  𝒫 cpw 4108  {csn 4125   cuni 4372  cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  ccom 5042   Fn wfn 5799  wf 5800  ontowfo 5802  cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  Grpcgrp 17245  invgcminusg 17246  -gcsg 17247  CMndccmn 18016  Abelcabl 18017  TopOnctopon 20518  TopSpctps 20519  Homeochmeo 21366  TopGrpctgp 21685   tsums ctsu 21739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-topgen 15927  df-mre 16069  df-mrc 16070  df-acs 16072  df-plusf 17064  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-abl 18019  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-ntr 20634  df-nei 20712  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-tmd 21686  df-tgp 21687  df-tsms 21740
This theorem is referenced by:  tsmsxp  21768
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