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Theorem tsmsxplem1 24161
Description: Lemma for tsmsxp 24163. (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))
21elin2d 4205 . . 3 (𝜑𝐾 ∈ Fin)
3 elfpw 9394 . . . . . . . 8 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝐾𝐴𝐾 ∈ Fin))
43simplbi 497 . . . . . . 7 (𝐾 ∈ (𝒫 𝐴 ∩ Fin) → 𝐾𝐴)
51, 4syl 17 . . . . . 6 (𝜑𝐾𝐴)
65sselda 3983 . . . . 5 ((𝜑𝑗𝐾) → 𝑗𝐴)
7 tsmsxp.b . . . . . 6 𝐵 = (Base‘𝐺)
8 tsmsxp.j . . . . . 6 𝐽 = (TopOpen‘𝐺)
9 eqid 2737 . . . . . 6 (𝒫 𝐶 ∩ Fin) = (𝒫 𝐶 ∩ Fin)
10 tsmsxp.g . . . . . . 7 (𝜑𝐺 ∈ CMnd)
1110adantr 480 . . . . . 6 ((𝜑𝑗𝐴) → 𝐺 ∈ CMnd)
12 tsmsxp.2 . . . . . . . 8 (𝜑𝐺 ∈ TopGrp)
13 tgptps 24088 . . . . . . . 8 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
1412, 13syl 17 . . . . . . 7 (𝜑𝐺 ∈ TopSp)
1514adantr 480 . . . . . 6 ((𝜑𝑗𝐴) → 𝐺 ∈ TopSp)
16 tsmsxp.c . . . . . . 7 (𝜑𝐶𝑊)
1716adantr 480 . . . . . 6 ((𝜑𝑗𝐴) → 𝐶𝑊)
18 tsmsxp.f . . . . . . . . 9 (𝜑𝐹:(𝐴 × 𝐶)⟶𝐵)
19 fovcdm 7603 . . . . . . . . 9 ((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
2018, 19syl3an1 1164 . . . . . . . 8 ((𝜑𝑗𝐴𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
21203expa 1119 . . . . . . 7 (((𝜑𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
2221fmpttd 7135 . . . . . 6 ((𝜑𝑗𝐴) → (𝑘𝐶 ↦ (𝑗𝐹𝑘)):𝐶𝐵)
23 tsmsxp.1 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ (𝐺 tsums (𝑘𝐶 ↦ (𝑗𝐹𝑘))))
24 df-ima 5698 . . . . . . . 8 ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿)
258, 7tgptopon 24090 . . . . . . . . . . . . 13 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵))
2612, 25syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝐵))
27 tsmsxp.l . . . . . . . . . . . 12 (𝜑𝐿𝐽)
28 toponss 22933 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝐵) ∧ 𝐿𝐽) → 𝐿𝐵)
2926, 27, 28syl2anc 584 . . . . . . . . . . 11 (𝜑𝐿𝐵)
3029adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝐴) → 𝐿𝐵)
3130resmptd 6058 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3231rneqd 5949 . . . . . . . 8 ((𝜑𝑗𝐴) → ran ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ↾ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
3324, 32eqtrid 2789 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) = ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
34 tsmsxp.h . . . . . . . . . . . . 13 (𝜑𝐻:𝐴𝐵)
3534ffvelcdmda 7104 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ 𝐵)
36 tsmsxp.p . . . . . . . . . . . . 13 + = (+g𝐺)
37 eqid 2737 . . . . . . . . . . . . 13 (invg𝐺) = (invg𝐺)
38 tsmsxp.m . . . . . . . . . . . . 13 = (-g𝐺)
397, 36, 37, 38grpsubval 19003 . . . . . . . . . . . 12 (((𝐻𝑗) ∈ 𝐵𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4035, 39sylan 580 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
4140mpteq2dva 5242 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
42 tgpgrp 24086 . . . . . . . . . . . . . 14 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
4312, 42syl 17 . . . . . . . . . . . . 13 (𝜑𝐺 ∈ Grp)
4443adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → 𝐺 ∈ Grp)
457, 37grpinvcl 19005 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
4644, 45sylan 580 . . . . . . . . . . 11 (((𝜑𝑗𝐴) ∧ 𝑔𝐵) → ((invg𝐺)‘𝑔) ∈ 𝐵)
477, 37grpinvf 19004 . . . . . . . . . . . . 13 (𝐺 ∈ Grp → (invg𝐺):𝐵𝐵)
4844, 47syl 17 . . . . . . . . . . . 12 ((𝜑𝑗𝐴) → (invg𝐺):𝐵𝐵)
4948feqmptd 6977 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (invg𝐺) = (𝑔𝐵 ↦ ((invg𝐺)‘𝑔)))
50 eqidd 2738 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)))
51 oveq2 7439 . . . . . . . . . . 11 (𝑦 = ((invg𝐺)‘𝑔) → ((𝐻𝑗) + 𝑦) = ((𝐻𝑗) + ((invg𝐺)‘𝑔)))
5246, 49, 50, 51fmptco 7149 . . . . . . . . . 10 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) = (𝑔𝐵 ↦ ((𝐻𝑗) + ((invg𝐺)‘𝑔))))
5341, 52eqtr4d 2780 . . . . . . . . 9 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) = ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)))
5412adantr 480 . . . . . . . . . . 11 ((𝜑𝑗𝐴) → 𝐺 ∈ TopGrp)
558, 37grpinvhmeo 24094 . . . . . . . . . . 11 (𝐺 ∈ TopGrp → (invg𝐺) ∈ (𝐽Homeo𝐽))
5654, 55syl 17 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (invg𝐺) ∈ (𝐽Homeo𝐽))
57 eqid 2737 . . . . . . . . . . . 12 (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) = (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦))
5857, 7, 36, 8tgplacthmeo 24111 . . . . . . . . . . 11 ((𝐺 ∈ TopGrp ∧ (𝐻𝑗) ∈ 𝐵) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
5954, 35, 58syl2anc 584 . . . . . . . . . 10 ((𝜑𝑗𝐴) → (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽))
60 hmeoco 23780 . . . . . . . . . 10 (((invg𝐺) ∈ (𝐽Homeo𝐽) ∧ (𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∈ (𝐽Homeo𝐽)) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6156, 59, 60syl2anc 584 . . . . . . . . 9 ((𝜑𝑗𝐴) → ((𝑦𝐵 ↦ ((𝐻𝑗) + 𝑦)) ∘ (invg𝐺)) ∈ (𝐽Homeo𝐽))
6253, 61eqeltrd 2841 . . . . . . . 8 ((𝜑𝑗𝐴) → (𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽))
6327adantr 480 . . . . . . . 8 ((𝜑𝑗𝐴) → 𝐿𝐽)
64 hmeoima 23773 . . . . . . . 8 (((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) ∈ (𝐽Homeo𝐽) ∧ 𝐿𝐽) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6562, 63, 64syl2anc 584 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝑔𝐵 ↦ ((𝐻𝑗) 𝑔)) “ 𝐿) ∈ 𝐽)
6633, 65eqeltrrd 2842 . . . . . 6 ((𝜑𝑗𝐴) → ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ∈ 𝐽)
67 tsmsxp.z . . . . . . . . 9 0 = (0g𝐺)
687, 67, 38grpsubid1 19043 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝐻𝑗) ∈ 𝐵) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
6944, 35, 68syl2anc 584 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐻𝑗) 0 ) = (𝐻𝑗))
70 tsmsxp.3 . . . . . . . . 9 (𝜑0𝐿)
7170adantr 480 . . . . . . . 8 ((𝜑𝑗𝐴) → 0𝐿)
72 ovex 7464 . . . . . . . 8 ((𝐻𝑗) 0 ) ∈ V
73 eqid 2737 . . . . . . . . 9 (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) = (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))
74 oveq2 7439 . . . . . . . . 9 (𝑔 = 0 → ((𝐻𝑗) 𝑔) = ((𝐻𝑗) 0 ))
7573, 74elrnmpt1s 5970 . . . . . . . 8 (( 0𝐿 ∧ ((𝐻𝑗) 0 ) ∈ V) → ((𝐻𝑗) 0 ) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
7671, 72, 75sylancl 586 . . . . . . 7 ((𝜑𝑗𝐴) → ((𝐻𝑗) 0 ) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
7769, 76eqeltrrd 2842 . . . . . 6 ((𝜑𝑗𝐴) → (𝐻𝑗) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))
787, 8, 9, 11, 15, 17, 22, 23, 66, 77tsmsi 24142 . . . . 5 ((𝜑𝑗𝐴) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
796, 78syldan 591 . . . 4 ((𝜑𝑗𝐾) → ∃𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
8079ralrimiva 3146 . . 3 (𝜑 → ∀𝑗𝐾𝑦 ∈ (𝒫 𝐶 ∩ Fin)∀𝑧 ∈ (𝒫 𝐶 ∩ Fin)(𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
81 sseq1 4009 . . . . . 6 (𝑦 = (𝑓𝑗) → (𝑦𝑧 ↔ (𝑓𝑗) ⊆ 𝑧))
8281imbi1d 341 . . . . 5 (𝑦 = (𝑓𝑗) → ((𝑦𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) ↔ ((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))))
8382ralbidv 3178 . . . 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)))))
8483ac6sfi 9320 . . 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)))))
852, 80, 84syl2anc 584 . 2 (𝜑 → ∃𝑓(𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) ∧ ∀𝑗𝐾𝑧 ∈ (𝒫 𝐶 ∩ Fin)((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)))))
86 frn 6743 . . . . . . . . 9 (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
8786adantl 481 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ (𝒫 𝐶 ∩ Fin))
88 inss1 4237 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ 𝒫 𝐶
8987, 88sstrdi 3996 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ 𝒫 𝐶)
90 sspwuni 5100 . . . . . . 7 (ran 𝑓 ⊆ 𝒫 𝐶 ran 𝑓𝐶)
9189, 90sylib 218 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓𝐶)
92 tsmsxp.d . . . . . . . . 9 (𝜑𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin))
93 elfpw 9394 . . . . . . . . . 10 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) ↔ (𝐷 ⊆ (𝐴 × 𝐶) ∧ 𝐷 ∈ Fin))
9493simplbi 497 . . . . . . . . 9 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ⊆ (𝐴 × 𝐶))
95 rnss 5950 . . . . . . . . 9 (𝐷 ⊆ (𝐴 × 𝐶) → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
9692, 94, 953syl 18 . . . . . . . 8 (𝜑 → ran 𝐷 ⊆ ran (𝐴 × 𝐶))
97 rnxpss 6192 . . . . . . . 8 ran (𝐴 × 𝐶) ⊆ 𝐶
9896, 97sstrdi 3996 . . . . . . 7 (𝜑 → ran 𝐷𝐶)
9998adantr 480 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷𝐶)
10091, 99unssd 4192 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
1012adantr 480 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐾 ∈ Fin)
102 ffn 6736 . . . . . . . . . 10 (𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin) → 𝑓 Fn 𝐾)
103102adantl 481 . . . . . . . . 9 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓 Fn 𝐾)
104 dffn4 6826 . . . . . . . . 9 (𝑓 Fn 𝐾𝑓:𝐾onto→ran 𝑓)
105103, 104sylib 218 . . . . . . . 8 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑓:𝐾onto→ran 𝑓)
106 fofi 9351 . . . . . . . 8 ((𝐾 ∈ Fin ∧ 𝑓:𝐾onto→ran 𝑓) → ran 𝑓 ∈ Fin)
107101, 105, 106syl2anc 584 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
108 inss2 4238 . . . . . . . 8 (𝒫 𝐶 ∩ Fin) ⊆ Fin
10987, 108sstrdi 3996 . . . . . . 7 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ⊆ Fin)
110 unifi 9384 . . . . . . 7 ((ran 𝑓 ∈ Fin ∧ ran 𝑓 ⊆ Fin) → ran 𝑓 ∈ Fin)
111107, 109, 110syl2anc 584 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝑓 ∈ Fin)
112 elinel2 4202 . . . . . . . 8 (𝐷 ∈ (𝒫 (𝐴 × 𝐶) ∩ Fin) → 𝐷 ∈ Fin)
113 rnfi 9380 . . . . . . . 8 (𝐷 ∈ Fin → ran 𝐷 ∈ Fin)
11492, 112, 1133syl 18 . . . . . . 7 (𝜑 → ran 𝐷 ∈ Fin)
115114adantr 480 . . . . . 6 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ran 𝐷 ∈ Fin)
116 unfi 9211 . . . . . 6 (( ran 𝑓 ∈ Fin ∧ ran 𝐷 ∈ Fin) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
117111, 115, 116syl2anc 584 . . . . 5 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
118 elfpw 9394 . . . . 5 (( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin) ↔ (( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶 ∧ ( ran 𝑓 ∪ ran 𝐷) ∈ Fin))
119100, 117, 118sylanbrc 583 . . . 4 ((𝜑𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
120119adantrr 717 . . 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))
121 ssun2 4179 . . . 4 ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷)
122121a1i 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))
123119adantlr 715 . . . . . . . . 9 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ (𝒫 𝐶 ∩ Fin))
124 fvssunirn 6939 . . . . . . . . . . . . . 14 (𝑓𝑗) ⊆ ran 𝑓
125 ssun1 4178 . . . . . . . . . . . . . 14 ran 𝑓 ⊆ ( ran 𝑓 ∪ ran 𝐷)
126124, 125sstri 3993 . . . . . . . . . . . . 13 (𝑓𝑗) ⊆ ( ran 𝑓 ∪ ran 𝐷)
127 id 22 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → 𝑧 = ( ran 𝑓 ∪ ran 𝐷))
128126, 127sseqtrrid 4027 . . . . . . . . . . . 12 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (𝑓𝑗) ⊆ 𝑧)
129 pm5.5 361 . . . . . . . . . . . 12 ((𝑓𝑗) ⊆ 𝑧 → (((𝑓𝑗) ⊆ 𝑧 → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))) ↔ (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔))))
130128, 129syl 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))))
131 reseq2 5992 . . . . . . . . . . . . 13 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧) = ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)))
132131oveq2d 7447 . . . . . . . . . . . 12 (𝑧 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ 𝑧)) = (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))))
133132eleq1d 2826 . . . . . . . . . . 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))))
134130, 133bitrd 279 . . . . . . . . . 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))))
135134rspcv 3618 . . . . . . . . 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))))
136123, 135syl 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))))
13710ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ CMnd)
138 cmnmnd 19815 . . . . . . . . . . . . 13 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
139137, 138syl 17 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐺 ∈ Mnd)
140 simplr 769 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐾)
141117adantlr 715 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ∈ Fin)
142100adantlr 715 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶)
143142sselda 3983 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → 𝑘𝐶)
14418adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝐾) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
145144, 6jca 511 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝐾) → (𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴))
146193expa 1119 . . . . . . . . . . . . . . . . 17 (((𝐹:(𝐴 × 𝐶)⟶𝐵𝑗𝐴) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
147145, 146sylan 580 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝐾) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
148147adantlr 715 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘𝐶) → (𝑗𝐹𝑘) ∈ 𝐵)
149143, 148syldan 591 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ 𝐵)
150149fmpttd 7135 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)):( ran 𝑓 ∪ ran 𝐷)⟶𝐵)
151 eqid 2737 . . . . . . . . . . . . . 14 (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))
152 ovexd 7466 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑗𝐹𝑘) ∈ V)
15367fvexi 6920 . . . . . . . . . . . . . . 15 0 ∈ V
154153a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 0 ∈ V)
155151, 141, 152, 154fsuppmptdm 9416 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)) finSupp 0 )
1567, 67, 137, 141, 150, 155gsumcl 19933 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))) ∈ 𝐵)
157 velsn 4642 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ {𝑗} ↔ 𝑦 = 𝑗)
158 ovres 7599 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ {𝑗} ∧ 𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
159157, 158sylanbr 582 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑦𝐹𝑘))
160 oveq1 7438 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑗 → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
161160adantr 480 . . . . . . . . . . . . . . . 16 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦𝐹𝑘) = (𝑗𝐹𝑘))
162159, 161eqtrd 2777 . . . . . . . . . . . . . . 15 ((𝑦 = 𝑗𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷)) → (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘) = (𝑗𝐹𝑘))
163162mpteq2dva 5242 . . . . . . . . . . . . . 14 (𝑦 = 𝑗 → (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑦(𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))𝑘)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
164163oveq2d 7447 . . . . . . . . . . . . 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))))
1657, 164gsumsn 19972 . . . . . . . . . . . 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))))
166139, 140, 156, 165syl3anc 1373 . . . . . . . . . . 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))))
167 snfi 9083 . . . . . . . . . . . . 13 {𝑗} ∈ Fin
168167a1i 11 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ∈ Fin)
16918ad2antrr 726 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐹:(𝐴 × 𝐶)⟶𝐵)
1706adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝑗𝐴)
171170snssd 4809 . . . . . . . . . . . . . 14 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → {𝑗} ⊆ 𝐴)
172 xpss12 5700 . . . . . . . . . . . . . 14 (({𝑗} ⊆ 𝐴 ∧ ( ran 𝑓 ∪ ran 𝐷) ⊆ 𝐶) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
173171, 142, 172syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ⊆ (𝐴 × 𝐶))
174169, 173fssresd 6775 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))):({𝑗} × ( ran 𝑓 ∪ ran 𝐷))⟶𝐵)
175 xpfi 9358 . . . . . . . . . . . . . 14 (({𝑗} ∈ Fin ∧ ( ran 𝑓 ∪ ran 𝐷) ∈ Fin) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
176167, 141, 175sylancr 587 . . . . . . . . . . . . 13 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) ∈ Fin)
177174, 176, 154fdmfifsupp 9415 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) finSupp 0 )
1787, 67, 137, 168, 141, 174, 177gsumxp 19994 . . . . . . . . . . 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))))))
179142resmptd 6058 . . . . . . . . . . . 12 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷)) = (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘)))
180179oveq2d 7447 . . . . . . . . . . 11 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝑘 ∈ ( ran 𝑓 ∪ ran 𝐷) ↦ (𝑗𝐹𝑘))))
181166, 178, 1803eqtr4rd 2788 . . . . . . . . . 10 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → (𝐺 Σg ((𝑘𝐶 ↦ (𝑗𝐹𝑘)) ↾ ( ran 𝑓 ∪ ran 𝐷))) = (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))))
182181eleq1d 2826 . . . . . . . . 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))))
183 ovex 7464 . . . . . . . . . . 11 ((𝐻𝑗) 𝑔) ∈ V
18473, 183elrnmpti 5973 . . . . . . . . . 10 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) ∈ ran (𝑔𝐿 ↦ ((𝐻𝑗) 𝑔)) ↔ ∃𝑔𝐿 (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔))
185 isabl 19802 . . . . . . . . . . . . . . . 16 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ CMnd))
18643, 10, 185sylanbrc 583 . . . . . . . . . . . . . . 15 (𝜑𝐺 ∈ Abel)
187186ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝐺 ∈ Abel)
1886, 35syldan 591 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝐾) → (𝐻𝑗) ∈ 𝐵)
189188ad2antrr 726 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → (𝐻𝑗) ∈ 𝐵)
19029ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) → 𝐿𝐵)
191190sselda 3983 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐵)
1927, 38, 187, 189, 191ablnncan 19838 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) = 𝑔)
193 simpr 484 . . . . . . . . . . . . 13 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → 𝑔𝐿)
194192, 193eqeltrd 2841 . . . . . . . . . . . 12 ((((𝜑𝑗𝐾) ∧ 𝑓:𝐾⟶(𝒫 𝐶 ∩ Fin)) ∧ 𝑔𝐿) → ((𝐻𝑗) ((𝐻𝑗) 𝑔)) ∈ 𝐿)
195 oveq2 7439 . . . . . . . . . . . . 13 ((𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = ((𝐻𝑗) 𝑔) → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑗) ((𝐻𝑗) 𝑔)))
196195eleq1d 2826 . . . . . . . . . . . 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))
197194, 196syl5ibrcom 247 . . . . . . . . . . 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))
198197rexlimdva 3155 . . . . . . . . . 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))
199184, 198biimtrid 242 . . . . . . . . 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))
200182, 199sylbid 240 . . . . . . . 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))
201136, 200syld 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))
202201an32s 652 . . . . . 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))
203202ralimdva 3167 . . . . 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))
204203impr 454 . . . 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)
205 fveq2 6906 . . . . . . 7 (𝑗 = 𝑥 → (𝐻𝑗) = (𝐻𝑥))
206 sneq 4636 . . . . . . . . . 10 (𝑗 = 𝑥 → {𝑗} = {𝑥})
207206xpeq1d 5714 . . . . . . . . 9 (𝑗 = 𝑥 → ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
208207reseq2d 5997 . . . . . . . 8 (𝑗 = 𝑥 → (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
209208oveq2d 7447 . . . . . . 7 (𝑗 = 𝑥 → (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷)))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
210205, 209oveq12d 7449 . . . . . 6 (𝑗 = 𝑥 → ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
211210eleq1d 2826 . . . . 5 (𝑗 = 𝑥 → (((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
212211cbvralvw 3237 . . . 4 (∀𝑗𝐾 ((𝐻𝑗) (𝐺 Σg (𝐹 ↾ ({𝑗} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿 ↔ ∀𝑥𝐾 ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿)
213204, 212sylib 218 . . 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)
214 sseq2 4010 . . . . 5 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (ran 𝐷𝑛 ↔ ran 𝐷 ⊆ ( ran 𝑓 ∪ ran 𝐷)))
215 xpeq2 5706 . . . . . . . . . 10 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ({𝑥} × 𝑛) = ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))
216215reseq2d 5997 . . . . . . . . 9 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐹 ↾ ({𝑥} × 𝑛)) = (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))
217216oveq2d 7447 . . . . . . . 8 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛))) = (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷)))))
218217oveq2d 7447 . . . . . . 7 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) = ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))))
219218eleq1d 2826 . . . . . 6 (𝑛 = ( ran 𝑓 ∪ ran 𝐷) → (((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × 𝑛)))) ∈ 𝐿 ↔ ((𝐻𝑥) (𝐺 Σg (𝐹 ↾ ({𝑥} × ( ran 𝑓 ∪ ran 𝐷))))) ∈ 𝐿))
220219ralbidv 3178 . . . . 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))
221214, 220anbi12d 632 . . . 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)))
222221rspcev 3622 . . 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))
223120, 122, 213, 222syl12anc 837 . 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))
22485, 223exlimddv 1935 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 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cun 3949  cin 3950  wss 3951  𝒫 cpw 4600  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  Fincfn 8985  Basecbs 17247  +gcplusg 17297  TopOpenctopn 17466  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747  Grpcgrp 18951  invgcminusg 18952  -gcsg 18953  CMndccmn 19798  Abelcabl 19799  TopOnctopon 22916  TopSpctps 22938  Homeochmeo 23761  TopGrpctgp 24079   tsums ctsu 24134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-topgen 17488  df-mre 17629  df-mrc 17630  df-acs 17632  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-abl 19801  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-ntr 23028  df-nei 23106  df-cn 23235  df-cnp 23236  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-tmd 24080  df-tgp 24081  df-tsms 24135
This theorem is referenced by:  tsmsxp  24163
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