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Theorem tmdgsum 21946
Description: In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when 𝐴 is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
Hypotheses
Ref Expression
tmdgsum.j 𝐽 = (TopOpen‘𝐺)
tmdgsum.b 𝐵 = (Base‘𝐺)
Assertion
Ref Expression
tmdgsum ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝐵   𝑥,𝐺

Proof of Theorem tmdgsum
Dummy variables 𝑘 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . . . . . . 8 (𝑤 = ∅ → (𝐵𝑚 𝑤) = (𝐵𝑚 ∅))
21mpteq1d 4771 . . . . . . 7 (𝑤 = ∅ → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)))
3 xpeq1 5157 . . . . . . . . . 10 (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4 0xp 5233 . . . . . . . . . 10 (∅ × {𝐽}) = ∅
53, 4syl6eq 2701 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
65fveq2d 6233 . . . . . . . 8 (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
76oveq1d 6705 . . . . . . 7 (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
82, 7eleq12d 2724 . . . . . 6 (𝑤 = ∅ → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽)))
98imbi2d 329 . . . . 5 (𝑤 = ∅ → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))))
10 oveq2 6698 . . . . . . . 8 (𝑤 = 𝑦 → (𝐵𝑚 𝑤) = (𝐵𝑚 𝑦))
1110mpteq1d 4771 . . . . . . 7 (𝑤 = 𝑦 → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)))
12 xpeq1 5157 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
1312fveq2d 6233 . . . . . . . 8 (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
1413oveq1d 6705 . . . . . . 7 (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
1511, 14eleq12d 2724 . . . . . 6 (𝑤 = 𝑦 → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)))
1615imbi2d 329 . . . . 5 (𝑤 = 𝑦 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))))
17 oveq2 6698 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵𝑚 𝑤) = (𝐵𝑚 (𝑦 ∪ {𝑧})))
1817mpteq1d 4771 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19 xpeq1 5157 . . . . . . . . 9 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
2019fveq2d 6233 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
2120oveq1d 6705 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
2218, 21eleq12d 2724 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
2322imbi2d 329 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
24 oveq2 6698 . . . . . . . 8 (𝑤 = 𝐴 → (𝐵𝑚 𝑤) = (𝐵𝑚 𝐴))
2524mpteq1d 4771 . . . . . . 7 (𝑤 = 𝐴 → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)))
26 xpeq1 5157 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
2726fveq2d 6233 . . . . . . . 8 (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
2827oveq1d 6705 . . . . . . 7 (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
2925, 28eleq12d 2724 . . . . . 6 (𝑤 = 𝐴 → ((𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
3029imbi2d 329 . . . . 5 (𝑤 = 𝐴 → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽)) ↔ ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))))
31 elmapfn 7922 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 ∅) → 𝑥 Fn ∅)
32 fn0 6049 . . . . . . . . . 10 (𝑥 Fn ∅ ↔ 𝑥 = ∅)
3331, 32sylib 208 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑚 ∅) → 𝑥 = ∅)
3433oveq2d 6706 . . . . . . . 8 (𝑥 ∈ (𝐵𝑚 ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35 eqid 2651 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
3635gsum0 17325 . . . . . . . 8 (𝐺 Σg ∅) = (0g𝐺)
3734, 36syl6eq 2701 . . . . . . 7 (𝑥 ∈ (𝐵𝑚 ∅) → (𝐺 Σg 𝑥) = (0g𝐺))
3837mpteq2ia 4773 . . . . . 6 (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 ∅) ↦ (0g𝐺))
39 0ex 4823 . . . . . . . 8 ∅ ∈ V
40 tmdgsum.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
41 tmdgsum.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4240, 41tmdtopon 21932 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4342adantl 481 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
444fveq2i 6232 . . . . . . . . . 10 (∏t‘(∅ × {𝐽})) = (∏t‘∅)
4544eqcomi 2660 . . . . . . . . 9 (∏t‘∅) = (∏t‘(∅ × {𝐽}))
4645pttoponconst 21448 . . . . . . . 8 ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵𝑚 ∅)))
4739, 43, 46sylancr 696 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵𝑚 ∅)))
48 tmdmnd 21926 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
4948adantl 481 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
5041, 35mndidcl 17355 . . . . . . . 8 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
5149, 50syl 17 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g𝐺) ∈ 𝐵)
5247, 43, 51cnmptc 21513 . . . . . 6 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (0g𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
5338, 52syl5eqel 2734 . . . . 5 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54 oveq2 6698 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
5554cbvmptv 4783 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56 eqid 2651 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
57 simpl1l 1132 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝐺 ∈ CMnd)
58 simp2l 1107 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ∈ Fin)
59 snfi 8079 . . . . . . . . . . . . . 14 {𝑧} ∈ Fin
60 unfi 8268 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
6158, 59, 60sylancl 695 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑦 ∪ {𝑧}) ∈ Fin)
6261adantr 480 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑦 ∪ {𝑧}) ∈ Fin)
63 elmapi 7921 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
6463adantl 481 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
65 fvexd 6241 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (0g𝐺) ∈ V)
6664, 62, 65fdmfifsupp 8326 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑤 finSupp (0g𝐺))
67 simpl2r 1135 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → ¬ 𝑧𝑦)
68 disjsn 4278 . . . . . . . . . . . . 13 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
6967, 68sylibr 224 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑦 ∩ {𝑧}) = ∅)
70 eqidd 2652 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
7141, 35, 56, 57, 62, 64, 66, 69, 70gsumsplit 18374 . . . . . . . . . . 11 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg 𝑤) = ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧}))))
7271mpteq2dva 4777 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤)) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))))
7355, 72syl5eq 2697 . . . . . . . . 9 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))))
74 simp1r 1106 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐺 ∈ TopMnd)
7574, 42syl 17 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ (TopOn‘𝐵))
76 eqid 2651 . . . . . . . . . . . 12 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
7776pttoponconst 21448 . . . . . . . . . . 11 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))))
7861, 75, 77syl2anc 694 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))))
79 toponuni 20767 . . . . . . . . . . . . . 14 ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐵𝑚 (𝑦 ∪ {𝑧})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
8078, 79syl 17 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝐵𝑚 (𝑦 ∪ {𝑧})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
8180mpteq1d 4771 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑦)) = (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑦)))
82 topontop 20766 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
8374, 42, 823syl 18 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ Top)
84 fconst6g 6132 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
8583, 84syl 17 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
86 ssun1 3809 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
8786a1i 11 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
88 eqid 2651 . . . . . . . . . . . . . 14 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89 xpssres 5469 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽}))
9086, 89ax-mp 5 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽})
9190eqcomi 2660 . . . . . . . . . . . . . . 15 (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
9291fveq2i 6232 . . . . . . . . . . . . . 14 (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
9388, 76, 92ptrescn 21490 . . . . . . . . . . . . 13 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
9461, 85, 87, 93syl3anc 1366 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
9581, 94eqeltrd 2730 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑦)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (∏t‘(𝑦 × {𝐽}))))
96 eqid 2651 . . . . . . . . . . . . 13 (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
9796pttoponconst 21448 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵𝑚 𝑦)))
9858, 75, 97syl2anc 694 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵𝑚 𝑦)))
99 simp3 1083 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
100 oveq2 6698 . . . . . . . . . . 11 (𝑥 = (𝑤𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤𝑦)))
10178, 95, 98, 99, 100cnmpt11 21514 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤𝑦))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
10264feqmptd 6288 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑤 = (𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)))
103102reseq1d 5427 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}))
104 ssun2 3810 . . . . . . . . . . . . . . . 16 {𝑧} ⊆ (𝑦 ∪ {𝑧})
105 resmpt 5484 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘)))
106104, 105ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))
107103, 106syl6eq 2701 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑤 ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘)))
108107oveq2d 6706 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑤 ↾ {𝑧})) = (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))))
109 cmnmnd 18254 . . . . . . . . . . . . . . 15 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
11057, 109syl 17 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝐺 ∈ Mnd)
111 vex 3234 . . . . . . . . . . . . . . 15 𝑧 ∈ V
112111a1i 11 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑧 ∈ V)
113 vsnid 4242 . . . . . . . . . . . . . . . 16 𝑧 ∈ {𝑧}
114 elun2 3814 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
115113, 114mp1i 13 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
11664, 115ffvelrnd 6400 . . . . . . . . . . . . . 14 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝑤𝑧) ∈ 𝐵)
117 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑤𝑘) = (𝑤𝑧))
11841, 117gsumsn 18400 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))) = (𝑤𝑧))
119110, 112, 116, 118syl3anc 1366 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))) = (𝑤𝑧))
120108, 119eqtrd 2685 . . . . . . . . . . . 12 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐺 Σg (𝑤 ↾ {𝑧})) = (𝑤𝑧))
121120mpteq2dva 4777 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤 ↾ {𝑧}))) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)))
12280mpteq1d 4771 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)) = (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑧)))
123113, 114mp1i 13 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
12488, 76ptpjcn 21462 . . . . . . . . . . . . . 14 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
12561, 85, 123, 124syl3anc 1366 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
126122, 125eqeltrd 2730 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)))
127 fvconst2g 6508 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
12883, 123, 127syl2anc 694 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
129128oveq2d 6706 . . . . . . . . . . . 12 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧)) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
130126, 129eleqtrd 2732 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝑤𝑧)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
131121, 130eqeltrd 2730 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg (𝑤 ↾ {𝑧}))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
13240, 56, 74, 78, 101, 131cnmpt1plusg 21938 . . . . . . . . 9 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ ((𝐺 Σg (𝑤𝑦))(+g𝐺)(𝐺 Σg (𝑤 ↾ {𝑧})))) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
13373, 132eqeltrd 2730 . . . . . . . 8 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
1341333expia 1286 . . . . . . 7 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽)))
135134expcom 450 . . . . . 6 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → ((𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
136135a2d 29 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))))
1379, 16, 23, 30, 53, 136findcard2s 8242 . . . 4 (𝐴 ∈ Fin → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
138137com12 32 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝐴 ∈ Fin → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
1391383impia 1280 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
14042, 82syl 17 . . . . 5 (𝐺 ∈ TopMnd → 𝐽 ∈ Top)
141 xkopt 21506 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
142140, 141sylan 487 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1431423adant1 1099 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
144143oveq1d 6705 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145139, 144eleqtrrd 2733 1 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   cuni 4468  cmpt 4762   × cxp 5141  cres 5145   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Fincfn 7997  Basecbs 15904  +gcplusg 15988  TopOpenctopn 16129  tcpt 16146  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  CMndccmn 18239  Topctop 20746  TopOnctopon 20763   Cn ccn 21076   ^ko cxko 21412  TopMndctmd 21921
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-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  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-rest 16130  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  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-mulg 17588  df-cntz 17796  df-cmn 18241  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cn 21079  df-cnp 21080  df-cmp 21238  df-tx 21413  df-xko 21414  df-tmd 21923
This theorem is referenced by:  tmdgsum2  21947
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