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Theorem ptpcon 30469
 Description: The topological product of a collection of path-connected spaces is path-connected. The proof uses the axiom of choice. (Contributed by Mario Carneiro, 17-Feb-2015.)
Assertion
Ref Expression
ptpcon ((𝐴𝑉𝐹:𝐴⟶PCon) → (∏t𝐹) ∈ PCon)

Proof of Theorem ptpcon
Dummy variables 𝑓 𝑥 𝑦 𝑔 𝑡 𝑧 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcontop 30461 . . . . 5 (𝑥 ∈ PCon → 𝑥 ∈ Top)
21ssriv 3572 . . . 4 PCon ⊆ Top
3 fss 5969 . . . 4 ((𝐹:𝐴⟶PCon ∧ PCon ⊆ Top) → 𝐹:𝐴⟶Top)
42, 3mpan2 703 . . 3 (𝐹:𝐴⟶PCon → 𝐹:𝐴⟶Top)
5 pttop 21195 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
64, 5sylan2 490 . 2 ((𝐴𝑉𝐹:𝐴⟶PCon) → (∏t𝐹) ∈ Top)
7 fvi 6165 . . . . . . . . . 10 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
87ad2antrr 758 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ( I ‘𝐴) = 𝐴)
98eleq2d 2673 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡𝐴))
109biimpa 500 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → 𝑡𝐴)
11 simplr 788 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝐹:𝐴⟶PCon)
1211ffvelrnda 6267 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝐹𝑡) ∈ PCon)
13 simprl 790 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 (∏t𝐹))
14 eqid 2610 . . . . . . . . . . . . . . . 16 (∏t𝐹) = (∏t𝐹)
1514ptuni 21207 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
164, 15sylan2 490 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝐴⟶PCon) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1716adantr 480 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1813, 17eleqtrrd 2691 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥X𝑡𝐴 (𝐹𝑡))
19 vex 3176 . . . . . . . . . . . . 13 𝑥 ∈ V
2019elixp 7801 . . . . . . . . . . . 12 (𝑥X𝑡𝐴 (𝐹𝑡) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡)))
2118, 20sylib 207 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑥 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡)))
2221simprd 478 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡))
2322r19.21bi 2916 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑥𝑡) ∈ (𝐹𝑡))
24 simprr 792 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 (∏t𝐹))
2524, 17eleqtrrd 2691 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦X𝑡𝐴 (𝐹𝑡))
26 vex 3176 . . . . . . . . . . . . 13 𝑦 ∈ V
2726elixp 7801 . . . . . . . . . . . 12 (𝑦X𝑡𝐴 (𝐹𝑡) ↔ (𝑦 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡)))
2825, 27sylib 207 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑦 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡)))
2928simprd 478 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡))
3029r19.21bi 2916 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑦𝑡) ∈ (𝐹𝑡))
31 eqid 2610 . . . . . . . . . 10 (𝐹𝑡) = (𝐹𝑡)
3231pconcn 30460 . . . . . . . . 9 (((𝐹𝑡) ∈ PCon ∧ (𝑥𝑡) ∈ (𝐹𝑡) ∧ (𝑦𝑡) ∈ (𝐹𝑡)) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
3312, 23, 30, 32syl3anc 1318 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
34 df-rex 2902 . . . . . . . 8 (∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)) ↔ ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3533, 34sylib 207 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3610, 35syldan 486 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3736ralrimiva 2949 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
38 fvex 6113 . . . . . 6 ( I ‘𝐴) ∈ V
39 eleq1 2676 . . . . . . 7 (𝑓 = (𝑔𝑡) → (𝑓 ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑡) ∈ (II Cn (𝐹𝑡))))
40 fveq1 6102 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘0) = ((𝑔𝑡)‘0))
4140eqeq1d 2612 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘0) = (𝑥𝑡) ↔ ((𝑔𝑡)‘0) = (𝑥𝑡)))
42 fveq1 6102 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘1) = ((𝑔𝑡)‘1))
4342eqeq1d 2612 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘1) = (𝑦𝑡) ↔ ((𝑔𝑡)‘1) = (𝑦𝑡)))
4441, 43anbi12d 743 . . . . . . 7 (𝑓 = (𝑔𝑡) → (((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)) ↔ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))
4539, 44anbi12d 743 . . . . . 6 (𝑓 = (𝑔𝑡) → ((𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))) ↔ ((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)))))
4638, 45ac6s2 9191 . . . . 5 (∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))) → ∃𝑔(𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)))))
4737, 46syl 17 . . . 4 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∃𝑔(𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)))))
48 iitopon 22490 . . . . . . 7 II ∈ (TopOn‘(0[,]1))
4948a1i 11 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → II ∈ (TopOn‘(0[,]1)))
50 simplll 794 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐴𝑉)
5111adantr 480 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐹:𝐴⟶PCon)
5251, 4syl 17 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐹:𝐴⟶Top)
538adantr 480 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ( I ‘𝐴) = 𝐴)
5453eleq2d 2673 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖 ∈ ( I ‘𝐴) ↔ 𝑖𝐴))
5554biimpar 501 . . . . . . . . . . 11 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → 𝑖 ∈ ( I ‘𝐴))
56 simprr 792 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))
57 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (𝑔𝑡) = (𝑔𝑖))
58 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝐹𝑡) = (𝐹𝑖))
5958oveq2d 6565 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (II Cn (𝐹𝑡)) = (II Cn (𝐹𝑖)))
6057, 59eleq12d 2682 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑖) ∈ (II Cn (𝐹𝑖))))
6157fveq1d 6105 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘0) = ((𝑔𝑖)‘0))
62 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑥𝑡) = (𝑥𝑖))
6361, 62eqeq12d 2625 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘0) = (𝑥𝑡) ↔ ((𝑔𝑖)‘0) = (𝑥𝑖)))
6457fveq1d 6105 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘1) = ((𝑔𝑖)‘1))
65 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑦𝑡) = (𝑦𝑖))
6664, 65eqeq12d 2625 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘1) = (𝑦𝑡) ↔ ((𝑔𝑖)‘1) = (𝑦𝑖)))
6763, 66anbi12d 743 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)) ↔ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
6860, 67anbi12d 743 . . . . . . . . . . . . 13 (𝑡 = 𝑖 → (((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))) ↔ ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖)))))
6968rspccva 3281 . . . . . . . . . . . 12 ((∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))) ∧ 𝑖 ∈ ( I ‘𝐴)) → ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
7056, 69sylan 487 . . . . . . . . . . 11 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖 ∈ ( I ‘𝐴)) → ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
7155, 70syldan 486 . . . . . . . . . 10 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
7271simpld 474 . . . . . . . . 9 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑔𝑖) ∈ (II Cn (𝐹𝑖)))
73 iiuni 22492 . . . . . . . . . 10 (0[,]1) = II
74 eqid 2610 . . . . . . . . . 10 (𝐹𝑖) = (𝐹𝑖)
7573, 74cnf 20860 . . . . . . . . 9 ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) → (𝑔𝑖):(0[,]1)⟶ (𝐹𝑖))
7672, 75syl 17 . . . . . . . 8 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑔𝑖):(0[,]1)⟶ (𝐹𝑖))
7776feqmptd 6159 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑔𝑖) = (𝑧 ∈ (0[,]1) ↦ ((𝑔𝑖)‘𝑧)))
7877, 72eqeltrrd 2689 . . . . . 6 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (𝑧 ∈ (0[,]1) ↦ ((𝑔𝑖)‘𝑧)) ∈ (II Cn (𝐹𝑖)))
7914, 49, 50, 52, 78ptcn 21240 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) ∈ (II Cn (∏t𝐹)))
8071simprd 478 . . . . . . . 8 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖)))
8180simpld 474 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → ((𝑔𝑖)‘0) = (𝑥𝑖))
8281mpteq2dva 4672 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) = (𝑖𝐴 ↦ (𝑥𝑖)))
83 0elunit 12161 . . . . . . 7 0 ∈ (0[,]1)
84 mptexg 6389 . . . . . . . 8 (𝐴𝑉 → (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V)
8550, 84syl 17 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V)
86 fveq2 6103 . . . . . . . . 9 (𝑧 = 0 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘0))
8786mpteq2dv 4673 . . . . . . . 8 (𝑧 = 0 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
88 eqid 2610 . . . . . . . 8 (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))
8987, 88fvmptg 6189 . . . . . . 7 ((0 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
9083, 85, 89sylancr 694 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
9121simpld 474 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 Fn 𝐴)
9291adantr 480 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑥 Fn 𝐴)
93 dffn5 6151 . . . . . . 7 (𝑥 Fn 𝐴𝑥 = (𝑖𝐴 ↦ (𝑥𝑖)))
9492, 93sylib 207 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑥 = (𝑖𝐴 ↦ (𝑥𝑖)))
9582, 90, 943eqtr4d 2654 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥)
9680simprd 478 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) ∧ 𝑖𝐴) → ((𝑔𝑖)‘1) = (𝑦𝑖))
9796mpteq2dva 4672 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) = (𝑖𝐴 ↦ (𝑦𝑖)))
98 1elunit 12162 . . . . . . 7 1 ∈ (0[,]1)
99 mptexg 6389 . . . . . . . 8 (𝐴𝑉 → (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V)
10050, 99syl 17 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V)
101 fveq2 6103 . . . . . . . . 9 (𝑧 = 1 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘1))
102101mpteq2dv 4673 . . . . . . . 8 (𝑧 = 1 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
103102, 88fvmptg 6189 . . . . . . 7 ((1 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
10498, 100, 103sylancr 694 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
10528simpld 474 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 Fn 𝐴)
106105adantr 480 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑦 Fn 𝐴)
107 dffn5 6151 . . . . . . 7 (𝑦 Fn 𝐴𝑦 = (𝑖𝐴 ↦ (𝑦𝑖)))
108106, 107sylib 207 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝑦 = (𝑖𝐴 ↦ (𝑦𝑖)))
10997, 104, 1083eqtr4d 2654 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦)
110 fveq1 6102 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0))
111110eqeq1d 2612 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘0) = 𝑥 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥))
112 fveq1 6102 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1))
113112eqeq1d 2612 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘1) = 𝑦 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦))
114111, 113anbi12d 743 . . . . . 6 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ (((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦)))
115114rspcev 3282 . . . . 5 (((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) ∈ (II Cn (∏t𝐹)) ∧ (((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦)) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
11679, 95, 109, 115syl12anc 1316 . . . 4 ((((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
11747, 116exlimddv 1850 . . 3 (((𝐴𝑉𝐹:𝐴⟶PCon) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
118117ralrimivva 2954 . 2 ((𝐴𝑉𝐹:𝐴⟶PCon) → ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
119 eqid 2610 . . 3 (∏t𝐹) = (∏t𝐹)
120119ispcon 30459 . 2 ((∏t𝐹) ∈ PCon ↔ ((∏t𝐹) ∈ Top ∧ ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
1216, 118, 120sylanbrc 695 1 ((𝐴𝑉𝐹:𝐴⟶PCon) → (∏t𝐹) ∈ PCon)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   ↦ cmpt 4643   I cid 4948   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Xcixp 7794  0cc0 9815  1c1 9816  [,]cicc 12049  ∏tcpt 15922  Topctop 20517  TopOnctopon 20518   Cn ccn 20838  IIcii 22486  PConcpcon 30455 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-reg 8380  ax-inf2 8421  ax-ac2 9168  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  ax-pre-sup 9893 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-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  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-fi 8200  df-sup 8231  df-inf 8232  df-r1 8510  df-rank 8511  df-card 8648  df-ac 8822  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-icc 12053  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-topgen 15927  df-pt 15928  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cn 20841  df-cnp 20842  df-ii 22488  df-pcon 30457 This theorem is referenced by: (None)
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