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Theorem tmdgsum 22622
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)) ∈ ((𝐽ko 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 7156 . . . . . . . 8 (𝑤 = ∅ → (𝐵m 𝑤) = (𝐵m ∅))
21mpteq1d 5152 . . . . . . 7 (𝑤 = ∅ → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)))
3 xpeq1 5568 . . . . . . . . . 10 (𝑤 = ∅ → (𝑤 × {𝐽}) = (∅ × {𝐽}))
4 0xp 5648 . . . . . . . . . 10 (∅ × {𝐽}) = ∅
53, 4syl6eq 2877 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 × {𝐽}) = ∅)
65fveq2d 6671 . . . . . . . 8 (𝑤 = ∅ → (∏t‘(𝑤 × {𝐽})) = (∏t‘∅))
76oveq1d 7163 . . . . . . 7 (𝑤 = ∅ → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘∅) Cn 𝐽))
82, 7eleq12d 2912 . . . . . 6 (𝑤 = ∅ → ((𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) ↔ (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽)))
98imbi2d 342 . . . . 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 7156 . . . . . . . 8 (𝑤 = 𝑦 → (𝐵m 𝑤) = (𝐵m 𝑦))
1110mpteq1d 5152 . . . . . . 7 (𝑤 = 𝑦 → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)))
12 xpeq1 5568 . . . . . . . . 9 (𝑤 = 𝑦 → (𝑤 × {𝐽}) = (𝑦 × {𝐽}))
1312fveq2d 6671 . . . . . . . 8 (𝑤 = 𝑦 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝑦 × {𝐽})))
1413oveq1d 7163 . . . . . . 7 (𝑤 = 𝑦 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝑦 × {𝐽})) Cn 𝐽))
1511, 14eleq12d 2912 . . . . . 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 342 . . . . 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 7156 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (𝐵m 𝑤) = (𝐵m (𝑦 ∪ {𝑧})))
1817mpteq1d 5152 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)))
19 xpeq1 5568 . . . . . . . . 9 (𝑤 = (𝑦 ∪ {𝑧}) → (𝑤 × {𝐽}) = ((𝑦 ∪ {𝑧}) × {𝐽}))
2019fveq2d 6671 . . . . . . . 8 (𝑤 = (𝑦 ∪ {𝑧}) → (∏t‘(𝑤 × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})))
2120oveq1d 7163 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) Cn 𝐽))
2218, 21eleq12d 2912 . . . . . 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 342 . . . . 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 7156 . . . . . . . 8 (𝑤 = 𝐴 → (𝐵m 𝑤) = (𝐵m 𝐴))
2524mpteq1d 5152 . . . . . . 7 (𝑤 = 𝐴 → (𝑥 ∈ (𝐵m 𝑤) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑥)))
26 xpeq1 5568 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤 × {𝐽}) = (𝐴 × {𝐽}))
2726fveq2d 6671 . . . . . . . 8 (𝑤 = 𝐴 → (∏t‘(𝑤 × {𝐽})) = (∏t‘(𝐴 × {𝐽})))
2827oveq1d 7163 . . . . . . 7 (𝑤 = 𝐴 → ((∏t‘(𝑤 × {𝐽})) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
2925, 28eleq12d 2912 . . . . . 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 342 . . . . 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 8419 . . . . . . . . . 10 (𝑥 ∈ (𝐵m ∅) → 𝑥 Fn ∅)
32 fn0 6476 . . . . . . . . . 10 (𝑥 Fn ∅ ↔ 𝑥 = ∅)
3331, 32sylib 219 . . . . . . . . 9 (𝑥 ∈ (𝐵m ∅) → 𝑥 = ∅)
3433oveq2d 7164 . . . . . . . 8 (𝑥 ∈ (𝐵m ∅) → (𝐺 Σg 𝑥) = (𝐺 Σg ∅))
35 eqid 2826 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
3635gsum0 17883 . . . . . . . 8 (𝐺 Σg ∅) = (0g𝐺)
3734, 36syl6eq 2877 . . . . . . 7 (𝑥 ∈ (𝐵m ∅) → (𝐺 Σg 𝑥) = (0g𝐺))
3837mpteq2ia 5154 . . . . . 6 (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)) = (𝑥 ∈ (𝐵m ∅) ↦ (0g𝐺))
39 0ex 5208 . . . . . . . 8 ∅ ∈ V
40 tmdgsum.j . . . . . . . . . 10 𝐽 = (TopOpen‘𝐺)
41 tmdgsum.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
4240, 41tmdtopon 22608 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐽 ∈ (TopOn‘𝐵))
4342adantl 482 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐽 ∈ (TopOn‘𝐵))
444fveq2i 6670 . . . . . . . . . 10 (∏t‘(∅ × {𝐽})) = (∏t‘∅)
4544eqcomi 2835 . . . . . . . . 9 (∏t‘∅) = (∏t‘(∅ × {𝐽}))
4645pttoponconst 22124 . . . . . . . 8 ((∅ ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘∅) ∈ (TopOn‘(𝐵m ∅)))
4739, 43, 46sylancr 587 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (∏t‘∅) ∈ (TopOn‘(𝐵m ∅)))
48 tmdmnd 22602 . . . . . . . . 9 (𝐺 ∈ TopMnd → 𝐺 ∈ Mnd)
4948adantl 482 . . . . . . . 8 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → 𝐺 ∈ Mnd)
5041, 35mndidcl 17914 . . . . . . . 8 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
5149, 50syl 17 . . . . . . 7 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (0g𝐺) ∈ 𝐵)
5247, 43, 51cnmptc 22189 . . . . . 6 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵m ∅) ↦ (0g𝐺)) ∈ ((∏t‘∅) Cn 𝐽))
5338, 52eqeltrid 2922 . . . . 5 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) → (𝑥 ∈ (𝐵m ∅) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘∅) Cn 𝐽))
54 oveq2 7156 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑤))
5554cbvmptv 5166 . . . . . . . . . 10 (𝑥 ∈ (𝐵m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑥)) = (𝑤 ∈ (𝐵m (𝑦 ∪ {𝑧})) ↦ (𝐺 Σg 𝑤))
56 eqid 2826 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
57 simpl1l 1218 . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . 14 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → 𝑦 ∈ Fin)
59 snfi 8583 . . . . . . . . . . . . . 14 {𝑧} ∈ Fin
60 unfi 8774 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
6158, 59, 60sylancl 586 . . . . . . . . . . . . 13 (((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd) ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑥 ∈ (𝐵m 𝑦) ↦ (𝐺 Σg 𝑥)) ∈ ((∏t‘(𝑦 × {𝐽})) Cn 𝐽)) → (𝑦 ∪ {𝑧}) ∈ Fin)
6261adantr 481 . . . . . . . . . . . 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 8418 . . . . . . . . . . . . 13 (𝑤 ∈ (𝐵m (𝑦 ∪ {𝑧})) → 𝑤:(𝑦 ∪ {𝑧})⟶𝐵)
6463adantl 482 . . . . . . . . . . . 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 6682 . . . . . . . . . . . . 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 8832 . . . . . . . . . . . 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 1221 . . . . . . . . . . . . 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 4646 . . . . . . . . . . . . 13 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
6967, 68sylibr 235 . . . . . . . . . . . 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 2827 . . . . . . . . . . . 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 18968 . . . . . . . . . . 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 5158 . . . . . . . . . 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 2873 . . . . . . . . 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 1192 . . . . . . . . . 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 2826 . . . . . . . . . . . 12 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
7776pttoponconst 22124 . . . . . . . . . . 11 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) ∈ (TopOn‘(𝐵m (𝑦 ∪ {𝑧}))))
7861, 75, 77syl2anc 584 . . . . . . . . . 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 21441 . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . 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 21440 . . . . . . . . . . . . . . 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 6565 . . . . . . . . . . . . . 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 4152 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . 14 (∏t‘((𝑦 ∪ {𝑧}) × {𝐽})) = (∏t‘((𝑦 ∪ {𝑧}) × {𝐽}))
89 xpssres 5888 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽}))
9086, 89ax-mp 5 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦) = (𝑦 × {𝐽})
9190eqcomi 2835 . . . . . . . . . . . . . . 15 (𝑦 × {𝐽}) = (((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦)
9291fveq2i 6670 . . . . . . . . . . . . . 14 (∏t‘(𝑦 × {𝐽})) = (∏t‘(((𝑦 ∪ {𝑧}) × {𝐽}) ↾ 𝑦))
9388, 76, 92ptrescn 22166 . . . . . . . . . . . . 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 1365 . . . . . . . . . . . 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 2918 . . . . . . . . . . 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 2826 . . . . . . . . . . . . 13 (∏t‘(𝑦 × {𝐽})) = (∏t‘(𝑦 × {𝐽}))
9796pttoponconst 22124 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ 𝐽 ∈ (TopOn‘𝐵)) → (∏t‘(𝑦 × {𝐽})) ∈ (TopOn‘(𝐵m 𝑦)))
9858, 75, 97syl2anc 584 . . . . . . . . . . 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 1132 . . . . . . . . . . 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 7156 . . . . . . . . . . 11 (𝑥 = (𝑤𝑦) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑤𝑦)))
10178, 95, 98, 99, 100cnmpt11 22190 . . . . . . . . . 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 6730 . . . . . . . . . . . . . . . 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 5851 . . . . . . . . . . . . . . 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 4153 . . . . . . . . . . . . . . . 16 {𝑧} ⊆ (𝑦 ∪ {𝑧})
105 resmpt 5904 . . . . . . . . . . . . . . . 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 2877 . . . . . . . . . . . . . 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 7164 . . . . . . . . . . . . 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 18842 . . . . . . . . . . . . . . 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 3503 . . . . . . . . . . . . . . 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 4599 . . . . . . . . . . . . . . . 16 𝑧 ∈ {𝑧}
114 elun2 4157 . . . . . . . . . . . . . . . 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 6848 . . . . . . . . . . . . . 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 6667 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝑤𝑘) = (𝑤𝑧))
11841, 117gsumsn 18994 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑧 ∈ V ∧ (𝑤𝑧) ∈ 𝐵) → (𝐺 Σg (𝑘 ∈ {𝑧} ↦ (𝑤𝑘))) = (𝑤𝑧))
119110, 112, 116, 118syl3anc 1365 . . . . . . . . . . . . 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 2861 . . . . . . . . . . . 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 5158 . . . . . . . . . . 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 5152 . . . . . . . . . . . . 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 22138 . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . 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 2918 . . . . . . . . . . . 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 6960 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → (((𝑦 ∪ {𝑧}) × {𝐽})‘𝑧) = 𝐽)
12883, 123, 127syl2anc 584 . . . . . . . . . . . . 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 7164 . . . . . . . . . . . 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 2920 . . . . . . . . . . 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 2918 . . . . . . . . . 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 22614 . . . . . . . . 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 2918 . . . . . . . 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 1115 . . . . . . 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 414 . . . . . 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 8748 . . . 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 1111 . 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 22182 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
142140, 141sylan 580 . . . 4 ((𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
1431423adant1 1124 . . 3 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝐽ko 𝒫 𝐴) = (∏t‘(𝐴 × {𝐽})))
144143oveq1d 7163 . 2 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → ((𝐽ko 𝒫 𝐴) Cn 𝐽) = ((∏t‘(𝐴 × {𝐽})) Cn 𝐽))
145139, 144eleqtrrd 2921 1 ((𝐺 ∈ CMnd ∧ 𝐺 ∈ TopMnd ∧ 𝐴 ∈ Fin) → (𝑥 ∈ (𝐵m 𝐴) ↦ (𝐺 Σg 𝑥)) ∈ ((𝐽ko 𝒫 𝐴) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  Vcvv 3500  cun 3938  cin 3939  wss 3940  c0 4295  𝒫 cpw 4542  {csn 4564   cuni 4837  cmpt 5143   × cxp 5552  cres 5556   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  m cmap 8396  Fincfn 8498  Basecbs 16473  +gcplusg 16555  TopOpenctopn 16685  tcpt 16702  0gc0g 16703   Σg cgsu 16704  Mndcmnd 17900  CMndccmn 18826  Topctop 21420  TopOnctopon 21437   Cn ccn 21751  ko cxko 22088  TopMndctmd 22597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7399  df-om 7569  df-1st 7680  df-2nd 7681  df-supp 7822  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-map 8398  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fsupp 8823  df-fi 8864  df-oi 8963  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-fzo 13024  df-seq 13360  df-hash 13681  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-rest 16686  df-0g 16705  df-gsum 16706  df-topgen 16707  df-pt 16708  df-mre 16847  df-mrc 16848  df-acs 16850  df-plusf 17841  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-submnd 17945  df-mulg 18155  df-cntz 18377  df-cmn 18828  df-top 21421  df-topon 21438  df-topsp 21460  df-bases 21473  df-cn 21754  df-cnp 21755  df-cmp 21914  df-tx 22089  df-xko 22090  df-tmd 22599
This theorem is referenced by:  tmdgsum2  22623
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