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Theorem tmdgsum 21709
 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 6557 . . . . . . . 8 (𝑤 = ∅ → (𝐵𝑚 𝑤) = (𝐵𝑚 ∅))
21mpteq1d 4666 . . . . . . 7 (𝑤 = ∅ → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)))
3 xpeq1 5052 . . . . . . . . . 10 (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4 0xp 5122 . . . . . . . . . 10 (∅ × {𝐽}) = ∅
53, 4syl6eq 2660 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
65fveq2d 6107 . . . . . . . 8 (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
76oveq1d 6564 . . . . . . 7 (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
82, 7eleq12d 2682 . . . . . 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 6557 . . . . . . . 8 (𝑤 = 𝑦 → (𝐵𝑚 𝑤) = (𝐵𝑚 𝑦))
1110mpteq1d 4666 . . . . . . 7 (𝑤 = 𝑦 → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)))
12 xpeq1 5052 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
1312fveq2d 6107 . . . . . . . 8 (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
1413oveq1d 6564 . . . . . . 7 (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
1511, 14eleq12d 2682 . . . . . 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 6557 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵𝑚 𝑤) = (𝐵𝑚 (𝑦 ∪ {𝑧})))
1817mpteq1d 4666 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19 xpeq1 5052 . . . . . . . . 9 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
2019fveq2d 6107 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
2120oveq1d 6564 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
2218, 21eleq12d 2682 . . . . . 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 6557 . . . . . . . 8 (𝑤 = 𝐴 → (𝐵𝑚 𝑤) = (𝐵𝑚 𝐴))
2524mpteq1d 4666 . . . . . . 7 (𝑤 = 𝐴 → (𝑥 ∈ (𝐵𝑚 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)))
26 xpeq1 5052 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
2726fveq2d 6107 . . . . . . . 8 (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
2827oveq1d 6564 . . . . . . 7 (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
2925, 28eleq12d 2682 . . . . . 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 7766 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 ∅) → 𝑥 Fn ∅)
32 fn0 5924 . . . . . . . . . 10 (𝑥 Fn ∅ ↔ 𝑥 = ∅)
3331, 32sylib 207 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑚 ∅) → 𝑥 = ∅)
3433oveq2d 6565 . . . . . . . 8 (𝑥 ∈ (𝐵𝑚 ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35 eqid 2610 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
3635gsum0 17101 . . . . . . . 8 (𝐺 Σg ∅) = (0g𝐺)
3734, 36syl6eq 2660 . . . . . . 7 (𝑥 ∈ (𝐵𝑚 ∅) → (𝐺 Σg 𝑥) = (0g𝐺))
3837mpteq2ia 4668 . . . . . 6 (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵𝑚 ∅) ↦ (0g𝐺))
39 0ex 4718 . . . . . . . 8 ∅ ∈ V
40 tmdgsum.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
41 tmdgsum.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4240, 41tmdtopon 21695 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4342adantl 481 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
444fveq2i 6106 . . . . . . . . . 10 (∏t‘(∅ × {𝐽})) = (∏t‘∅)
4544eqcomi 2619 . . . . . . . . 9 (∏t‘∅) = (∏t‘(∅ × {𝐽}))
4645pttoponconst 21210 . . . . . . . 8 ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵𝑚 ∅)))
4739, 43, 46sylancr 694 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵𝑚 ∅)))
48 tmdmnd 21689 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
4948adantl 481 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
5041, 35mndidcl 17131 . . . . . . . 8 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
5149, 50syl 17 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g𝐺) ∈ 𝐵)
5247, 43, 51cnmptc 21275 . . . . . 6 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (0g𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
5338, 52syl5eqel 2692 . . . . 5 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54 oveq2 6557 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
5554cbvmptv 4678 . . . . . . . . . 10 (𝑥 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56 eqid 2610 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
57 simpl1l 1105 . . . . . . . . . . . 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 1080 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ∈ Fin)
59 snfi 7923 . . . . . . . . . . . . . 14 {𝑧} ∈ Fin
60 unfi 8112 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
6158, 59, 60sylancl 693 . . . . . . . . . . . . 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 7765 . . . . . . . . . . . . 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 fvex 6113 . . . . . . . . . . . . . 14 (0g𝐺) ∈ V
6665a1i 11 . . . . . . . . . . . . 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)
6764, 62, 66fdmfifsupp 8168 . . . . . . . . . . . 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))
68 simpl2r 1108 . . . . . . . . . . . . 13 ((((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) ∧ 𝑤 ∈ (𝐵𝑚 (𝑦 ∪ {𝑧}))) → ¬ 𝑧𝑦)
69 disjsn 4192 . . . . . . . . . . . . 13 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
7068, 69sylibr 223 . . . . . . . . . . . 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}) = ∅)
71 eqidd 2611 . . . . . . . . . . . 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}))
7241, 35, 56, 57, 62, 64, 67, 70, 71gsumsplit 18151 . . . . . . . . . . 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}))))
7372mpteq2dva 4672 . . . . . . . . . 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})))))
7455, 73syl5eq 2656 . . . . . . . . 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})))))
75 simp1r 1079 . . . . . . . . . 10 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐺 ∈ TopMnd)
7675, 42syl 17 . . . . . . . . . . 11 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ (TopOn‘𝐵))
77 eqid 2610 . . . . . . . . . . . 12 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
7877pttoponconst 21210 . . . . . . . . . . 11 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))))
7961, 76, 78syl2anc 691 . . . . . . . . . 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}))))
80 toponuni 20542 . . . . . . . . . . . . . 14 ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵𝑚 (𝑦 ∪ {𝑧}))) → (𝐵𝑚 (𝑦 ∪ {𝑧})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
8179, 80syl 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})))
8281mpteq1d 4666 . . . . . . . . . . . 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)))
83 topontop 20541 . . . . . . . . . . . . . . 15 (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top)
8475, 42, 833syl 18 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝐽 ∈ Top)
85 fconst6g 6007 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → ((𝑦 ∪ {𝑧}) × {𝐽}):(𝑦 ∪ {𝑧})⟶Top)
8684, 85syl 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)
87 ssun1 3738 . . . . . . . . . . . . . 14 𝑦 ⊆ (𝑦 ∪ {𝑧})
8887a1i 11 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ⊆ (𝑦 ∪ {𝑧}))
89 eqid 2610 . . . . . . . . . . . . . 14 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
90 xpssres 5354 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽}))
9187, 90ax-mp 5 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽})
9291eqcomi 2619 . . . . . . . . . . . . . . 15 (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
9392fveq2i 6106 . . . . . . . . . . . . . 14 (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
9489, 77, 93ptrescn 21252 . . . . . . . . . . . . 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}))))
9561, 86, 88, 94syl3anc 1318 . . . . . . . . . . . 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}))))
9682, 95eqeltrd 2688 . . . . . . . . . . 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}))))
97 eqid 2610 . . . . . . . . . . . . 13 (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
9897pttoponconst 21210 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵𝑚 𝑦)))
9958, 76, 98syl2anc 691 . . . . . . . . . . 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)))
100 simp3 1056 . . . . . . . . . . 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))
101 oveq2 6557 . . . . . . . . . . 11 (𝑥 = (𝑤𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤𝑦)))
10279, 96, 99, 100, 101cnmpt11 21276 . . . . . . . . . 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))
10364feqmptd 6159 . . . . . . . . . . . . . . . 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)))
104103reseq1d 5316 . . . . . . . . . . . . . . 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}))
105 ssun2 3739 . . . . . . . . . . . . . . . 16 {𝑧} ⊆ (𝑦 ∪ {𝑧})
106 resmpt 5369 . . . . . . . . . . . . . . . 16 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘)))
107105, 106ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑘 ∈ (𝑦 ∪ {𝑧}) ↦ (𝑤𝑘)) ↾ {𝑧}) = (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))
108104, 107syl6eq 2660 . . . . . . . . . . . . . 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)))
109108oveq2d 6565 . . . . . . . . . . . . 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))))
110 cmnmnd 18031 . . . . . . . . . . . . . . 15 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
11157, 110syl 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)
112 vex 3176 . . . . . . . . . . . . . . 15 𝑧 ∈ V
113112a1i 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)
114 vsnid 4156 . . . . . . . . . . . . . . . 16 𝑧 ∈ {𝑧}
115 elun2 3743 . . . . . . . . . . . . . . . 16 (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
116114, 115mp1i 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}))
11764, 116ffvelrnd 6268 . . . . . . . . . . . . . 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)
118 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑤𝑘) = (𝑤𝑧))
11941, 118gsumsn 18177 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))) = (𝑤𝑧))
120111, 113, 117, 119syl3anc 1318 . . . . . . . . . . . . 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))
121109, 120eqtrd 2644 . . . . . . . . . . . 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))
122121mpteq2dva 4672 . . . . . . . . . . 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)))
12381mpteq1d 4666 . . . . . . . . . . . . 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)))
124114, 115mp1i 13 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
12589, 77ptpjcn 21224 . . . . . . . . . . . . . 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)))
12661, 86, 124, 125syl3anc 1318 . . . . . . . . . . . . 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)))
127123, 126eqeltrd 2688 . . . . . . . . . . . 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)))
128 fvconst2g 6372 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
12984, 124, 128syl2anc 691 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵𝑚 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
130129oveq2d 6565 . . . . . . . . . . . 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))
131127, 130eleqtrd 2690 . . . . . . . . . . 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))
132122, 131eqeltrd 2688 . . . . . . . . . 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))
13340, 56, 75, 79, 102, 132cnmpt1plusg 21701 . . . . . . . . 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))
13474, 133eqeltrd 2688 . . . . . . . 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))
1351343expia 1259 . . . . . . 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)))
136135expcom 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))))
137136a2d 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))))
1389, 16, 23, 30, 53, 137findcard2s 8086 . . . 4 (𝐴 ∈ Fin → ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
139138com12 32 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝐴 ∈ Fin → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽)))
1401393impia 1253 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵𝑚 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
14142, 83syl 17 . . . . 5 (𝐺 ∈ TopMnd → 𝐽 ∈ Top)
142 xkopt 21268 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
143141, 142sylan 487 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1441433adant1 1072 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽 ^ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
145144oveq1d 6564 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽 ^ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
146140, 145eleqtrrd 2691 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 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Fincfn 7841  Basecbs 15695  +gcplusg 15768  TopOpenctopn 15905  ∏tcpt 15922  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  CMndccmn 18016  Topctop 20517  TopOnctopon 20518   Cn ccn 20838   ^ko cxko 21174  TopMndctmd 21684 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-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  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-rest 15906  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  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-mulg 17364  df-cntz 17573  df-cmn 18018  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-cmp 21000  df-tx 21175  df-xko 21176  df-tmd 21686 This theorem is referenced by:  tmdgsum2  21710
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