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Theorem ptcmpfi 22356
 Description: A topological product of finitely many compact spaces is compact. This weak version of Tychonoff's theorem does not require the axiom of choice. (Contributed by Mario Carneiro, 8-Feb-2015.)
Assertion
Ref Expression
ptcmpfi ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)

Proof of Theorem ptcmpfi
Dummy variables 𝑣 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 6513 . . . . 5 (𝐹:𝐴⟶Comp → 𝐹 Fn 𝐴)
2 fnresdm 6465 . . . . 5 (𝐹 Fn 𝐴 → (𝐹𝐴) = 𝐹)
31, 2syl 17 . . . 4 (𝐹:𝐴⟶Comp → (𝐹𝐴) = 𝐹)
43adantl 482 . . 3 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (𝐹𝐴) = 𝐹)
54fveq2d 6673 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) = (∏t𝐹))
6 ssid 3993 . . . 4 𝐴𝐴
7 sseq1 3996 . . . . . 6 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
8 reseq2 5847 . . . . . . . . . 10 (𝑥 = ∅ → (𝐹𝑥) = (𝐹 ↾ ∅))
9 res0 5856 . . . . . . . . . 10 (𝐹 ↾ ∅) = ∅
108, 9syl6eq 2877 . . . . . . . . 9 (𝑥 = ∅ → (𝐹𝑥) = ∅)
1110fveq2d 6673 . . . . . . . 8 (𝑥 = ∅ → (∏t‘(𝐹𝑥)) = (∏t‘∅))
1211eleq1d 2902 . . . . . . 7 (𝑥 = ∅ → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘∅) ∈ Comp))
1312imbi2d 342 . . . . . 6 (𝑥 = ∅ → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp)))
147, 13imbi12d 346 . . . . 5 (𝑥 = ∅ → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (∅ ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp))))
15 sseq1 3996 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
16 reseq2 5847 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
1716fveq2d 6673 . . . . . . . 8 (𝑥 = 𝑦 → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹𝑦)))
1817eleq1d 2902 . . . . . . 7 (𝑥 = 𝑦 → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹𝑦)) ∈ Comp))
1918imbi2d 342 . . . . . 6 (𝑥 = 𝑦 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)))
2015, 19imbi12d 346 . . . . 5 (𝑥 = 𝑦 → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp))))
21 sseq1 3996 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
22 reseq2 5847 . . . . . . . . 9 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧})))
2322fveq2d 6673 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
2423eleq1d 2902 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
2524imbi2d 342 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
2621, 25imbi12d 346 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
27 sseq1 3996 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
28 reseq2 5847 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
2928fveq2d 6673 . . . . . . . 8 (𝑥 = 𝐴 → (∏t‘(𝐹𝑥)) = (∏t‘(𝐹𝐴)))
3029eleq1d 2902 . . . . . . 7 (𝑥 = 𝐴 → ((∏t‘(𝐹𝑥)) ∈ Comp ↔ (∏t‘(𝐹𝐴)) ∈ Comp))
3130imbi2d 342 . . . . . 6 (𝑥 = 𝐴 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp) ↔ ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)))
3227, 31imbi12d 346 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑥)) ∈ Comp)) ↔ (𝐴𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp))))
33 0ex 5208 . . . . . . . . 9 ∅ ∈ V
34 f0 6559 . . . . . . . . 9 ∅:∅⟶Top
35 pttop 22125 . . . . . . . . 9 ((∅ ∈ V ∧ ∅:∅⟶Top) → (∏t‘∅) ∈ Top)
3633, 34, 35mp2an 688 . . . . . . . 8 (∏t‘∅) ∈ Top
37 eqid 2826 . . . . . . . . . . . . 13 (∏t‘∅) = (∏t‘∅)
3837ptuni 22137 . . . . . . . . . . . 12 ((∅ ∈ V ∧ ∅:∅⟶Top) → X𝑥 ∈ ∅ (∅‘𝑥) = (∏t‘∅))
3933, 34, 38mp2an 688 . . . . . . . . . . 11 X𝑥 ∈ ∅ (∅‘𝑥) = (∏t‘∅)
40 ixp0x 8484 . . . . . . . . . . . 12 X𝑥 ∈ ∅ (∅‘𝑥) = {∅}
41 snfi 8588 . . . . . . . . . . . 12 {∅} ∈ Fin
4240, 41eqeltri 2914 . . . . . . . . . . 11 X𝑥 ∈ ∅ (∅‘𝑥) ∈ Fin
4339, 42eqeltrri 2915 . . . . . . . . . 10 (∏t‘∅) ∈ Fin
44 pwfi 8813 . . . . . . . . . 10 ( (∏t‘∅) ∈ Fin ↔ 𝒫 (∏t‘∅) ∈ Fin)
4543, 44mpbi 231 . . . . . . . . 9 𝒫 (∏t‘∅) ∈ Fin
46 pwuni 4873 . . . . . . . . 9 (∏t‘∅) ⊆ 𝒫 (∏t‘∅)
47 ssfi 8732 . . . . . . . . 9 ((𝒫 (∏t‘∅) ∈ Fin ∧ (∏t‘∅) ⊆ 𝒫 (∏t‘∅)) → (∏t‘∅) ∈ Fin)
4845, 46, 47mp2an 688 . . . . . . . 8 (∏t‘∅) ∈ Fin
4936, 48elini 4174 . . . . . . 7 (∏t‘∅) ∈ (Top ∩ Fin)
50 fincmp 21936 . . . . . . 7 ((∏t‘∅) ∈ (Top ∩ Fin) → (∏t‘∅) ∈ Comp)
5149, 50ax-mp 5 . . . . . 6 (∏t‘∅) ∈ Comp
52512a1i 12 . . . . 5 (∅ ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘∅) ∈ Comp))
53 ssun1 4152 . . . . . . . . 9 𝑦 ⊆ (𝑦 ∪ {𝑧})
54 id 22 . . . . . . . . 9 ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
5553, 54sstrid 3982 . . . . . . . 8 ((𝑦 ∪ {𝑧}) ⊆ 𝐴𝑦𝐴)
5655imim1i 63 . . . . . . 7 ((𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)))
57 eqid 2826 . . . . . . . . . . . . . 14 (∏t‘(𝐹𝑦)) = (∏t‘(𝐹𝑦))
58 eqid 2826 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘(𝐹 ↾ {𝑧}))
59 eqid 2826 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) = (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧})))
60 resabs1 5882 . . . . . . . . . . . . . . . . 17 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹𝑦))
6153, 60ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦) = (𝐹𝑦)
6261eqcomi 2835 . . . . . . . . . . . . . . 15 (𝐹𝑦) = ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦)
6362fveq2i 6672 . . . . . . . . . . . . . 14 (∏t‘(𝐹𝑦)) = (∏t‘((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ 𝑦))
64 ssun2 4153 . . . . . . . . . . . . . . . . 17 {𝑧} ⊆ (𝑦 ∪ {𝑧})
65 resabs1 5882 . . . . . . . . . . . . . . . . 17 ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}) = (𝐹 ↾ {𝑧}))
6664, 65ax-mp 5 . . . . . . . . . . . . . . . 16 ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}) = (𝐹 ↾ {𝑧})
6766eqcomi 2835 . . . . . . . . . . . . . . 15 (𝐹 ↾ {𝑧}) = ((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧})
6867fveq2i 6672 . . . . . . . . . . . . . 14 (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘((𝐹 ↾ (𝑦 ∪ {𝑧})) ↾ {𝑧}))
69 eqid 2826 . . . . . . . . . . . . . 14 (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣)) = (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣))
70 vex 3503 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
71 snex 5328 . . . . . . . . . . . . . . . 16 {𝑧} ∈ V
7270, 71unex 7462 . . . . . . . . . . . . . . 15 (𝑦 ∪ {𝑧}) ∈ V
7372a1i 11 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ∈ V)
74 simplr 765 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹:𝐴⟶Comp)
75 cmptop 21938 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ Comp → 𝑥 ∈ Top)
7675ssriv 3975 . . . . . . . . . . . . . . . 16 Comp ⊆ Top
77 fss 6526 . . . . . . . . . . . . . . . 16 ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
7874, 76, 77sylancl 586 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹:𝐴⟶Top)
79 simprr 769 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
8078, 79fssresd 6544 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹 ↾ (𝑦 ∪ {𝑧})):(𝑦 ∪ {𝑧})⟶Top)
81 eqidd 2827 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
82 simprl 767 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧𝑦)
83 disjsn 4646 . . . . . . . . . . . . . . 15 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
8482, 83sylibr 235 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∩ {𝑧}) = ∅)
8557, 58, 59, 63, 68, 69, 73, 80, 81, 84ptunhmeo 22351 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣)) ∈ (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧})))Homeo(∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧})))))
86 hmphi 22320 . . . . . . . . . . . . 13 ((𝑢 (∏t‘(𝐹𝑦)), 𝑣 (∏t‘(𝐹 ↾ {𝑧})) ↦ (𝑢𝑣)) ∈ (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧})))Homeo(∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧})))) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
8785, 86syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))))
881ad2antlr 723 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝐹 Fn 𝐴)
8964, 79sstrid 3982 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴)
90 vex 3503 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
9190snss 4717 . . . . . . . . . . . . . . . . 17 (𝑧𝐴 ↔ {𝑧} ⊆ 𝐴)
9289, 91sylibr 235 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧𝐴)
93 fnressn 6918 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐴𝑧𝐴) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹𝑧)⟩})
9488, 92, 93syl2anc 584 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹 ↾ {𝑧}) = {⟨𝑧, (𝐹𝑧)⟩})
9594fveq2d 6673 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘(𝐹 ↾ {𝑧})) = (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
96 eqid 2826 . . . . . . . . . . . . . . . . 17 (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) = (∏t‘{⟨𝑧, (𝐹𝑧)⟩})
9790a1i 11 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ V)
9874, 92ffvelrnd 6850 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ Comp)
9976, 98sseldi 3969 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ Top)
100 toptopon2 21461 . . . . . . . . . . . . . . . . . 18 ((𝐹𝑧) ∈ Top ↔ (𝐹𝑧) ∈ (TopOn‘ (𝐹𝑧)))
10199, 100sylib 219 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ∈ (TopOn‘ (𝐹𝑧)))
10296, 97, 101pt1hmeo 22349 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑥 (𝐹𝑧) ↦ {⟨𝑧, 𝑥⟩}) ∈ ((𝐹𝑧)Homeo(∏t‘{⟨𝑧, (𝐹𝑧)⟩})))
103 hmphi 22320 . . . . . . . . . . . . . . . 16 ((𝑥 (𝐹𝑧) ↦ {⟨𝑧, 𝑥⟩}) ∈ ((𝐹𝑧)Homeo(∏t‘{⟨𝑧, (𝐹𝑧)⟩})) → (𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
104102, 103syl 17 . . . . . . . . . . . . . . 15 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}))
105 cmphmph 22331 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ≃ (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) → ((𝐹𝑧) ∈ Comp → (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) ∈ Comp))
106104, 98, 105sylc 65 . . . . . . . . . . . . . 14 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘{⟨𝑧, (𝐹𝑧)⟩}) ∈ Comp)
10795, 106eqeltrd 2918 . . . . . . . . . . . . 13 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (∏t‘(𝐹 ↾ {𝑧})) ∈ Comp)
108 txcmp 22186 . . . . . . . . . . . . . 14 (((∏t‘(𝐹𝑦)) ∈ Comp ∧ (∏t‘(𝐹 ↾ {𝑧})) ∈ Comp) → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp)
109108expcom 414 . . . . . . . . . . . . 13 ((∏t‘(𝐹 ↾ {𝑧})) ∈ Comp → ((∏t‘(𝐹𝑦)) ∈ Comp → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp))
110107, 109syl 17 . . . . . . . . . . . 12 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ∈ Comp → ((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp))
111 cmphmph 22331 . . . . . . . . . . . 12 (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ≃ (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) → (((∏t‘(𝐹𝑦)) ×t (∏t‘(𝐹 ↾ {𝑧}))) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
11287, 110, 111sylsyld 61 . . . . . . . . . . 11 (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) ∧ (¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ((∏t‘(𝐹𝑦)) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))
113112expcom 414 . . . . . . . . . 10 ((¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → ((∏t‘(𝐹𝑦)) ∈ Comp → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
114113a2d 29 . . . . . . . . 9 ((¬ 𝑧𝑦 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp)))
115114ex 413 . . . . . . . 8 𝑧𝑦 → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp) → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
116115a2d 29 . . . . . . 7 𝑧𝑦 → (((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
11756, 116syl5 34 . . . . . 6 𝑧𝑦 → ((𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
118117adantl 482 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝑦𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝑦)) ∈ Comp)) → ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹 ↾ (𝑦 ∪ {𝑧}))) ∈ Comp))))
11914, 20, 26, 32, 52, 118findcard2s 8753 . . . 4 (𝐴 ∈ Fin → (𝐴𝐴 → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)))
1206, 119mpi 20 . . 3 (𝐴 ∈ Fin → ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp))
121120anabsi5 665 . 2 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t‘(𝐹𝐴)) ∈ Comp)
1225, 121eqeltrrd 2919 1 ((𝐴 ∈ Fin ∧ 𝐹:𝐴⟶Comp) → (∏t𝐹) ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107  Vcvv 3500   ∪ cun 3938   ∩ cin 3939   ⊆ wss 3940  ∅c0 4295  𝒫 cpw 4542  {csn 4564  ⟨cop 4570  ∪ cuni 4837   class class class wbr 5063   ↦ cmpt 5143   ↾ cres 5556   Fn wfn 6349  ⟶wf 6350  ‘cfv 6354  (class class class)co 7150   ∈ cmpo 7152  Xcixp 8455  Fincfn 8503  ∏tcpt 16707  Topctop 21436  TopOnctopon 21453  Compccmp 21929   ×t ctx 22103  Homeochmeo 22296   ≃ chmph 22297 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 7455 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-ral 3148  df-rex 3149  df-reu 3150  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-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 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-map 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-topgen 16712  df-pt 16713  df-top 21437  df-topon 21454  df-bases 21489  df-cn 21770  df-cnp 21771  df-cmp 21930  df-tx 22105  df-hmeo 22298  df-hmph 22299 This theorem is referenced by:  poimirlem30  34808
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