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Theorem ptpconn 32484
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
ptpconn ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ PConn)

Proof of Theorem ptpconn
Dummy variables 𝑓 𝑥 𝑦 𝑔 𝑡 𝑧 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pconntop 32476 . . . . 5 (𝑥 ∈ PConn → 𝑥 ∈ Top)
21ssriv 3974 . . . 4 PConn ⊆ Top
3 fss 6530 . . . 4 ((𝐹:𝐴⟶PConn ∧ PConn ⊆ Top) → 𝐹:𝐴⟶Top)
42, 3mpan2 689 . . 3 (𝐹:𝐴⟶PConn → 𝐹:𝐴⟶Top)
5 pttop 22193 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) ∈ Top)
64, 5sylan2 594 . 2 ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ Top)
7 fvi 6743 . . . . . . . . . 10 (𝐴𝑉 → ( I ‘𝐴) = 𝐴)
87ad2antrr 724 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ( I ‘𝐴) = 𝐴)
98eleq2d 2901 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑡 ∈ ( I ‘𝐴) ↔ 𝑡𝐴))
109biimpa 479 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → 𝑡𝐴)
11 simplr 767 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝐹:𝐴⟶PConn)
1211ffvelrnda 6854 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝐹𝑡) ∈ PConn)
13 simprl 769 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 (∏t𝐹))
14 eqid 2824 . . . . . . . . . . . . . . . 16 (∏t𝐹) = (∏t𝐹)
1514ptuni 22205 . . . . . . . . . . . . . . 15 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
164, 15sylan2 594 . . . . . . . . . . . . . 14 ((𝐴𝑉𝐹:𝐴⟶PConn) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1716adantr 483 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → X𝑡𝐴 (𝐹𝑡) = (∏t𝐹))
1813, 17eleqtrrd 2919 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥X𝑡𝐴 (𝐹𝑡))
19 vex 3500 . . . . . . . . . . . . 13 𝑥 ∈ V
2019elixp 8471 . . . . . . . . . . . 12 (𝑥X𝑡𝐴 (𝐹𝑡) ↔ (𝑥 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡)))
2118, 20sylib 220 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑥 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡)))
2221simprd 498 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑥𝑡) ∈ (𝐹𝑡))
2322r19.21bi 3211 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑥𝑡) ∈ (𝐹𝑡))
24 simprr 771 . . . . . . . . . . . . 13 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 (∏t𝐹))
2524, 17eleqtrrd 2919 . . . . . . . . . . . 12 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦X𝑡𝐴 (𝐹𝑡))
26 vex 3500 . . . . . . . . . . . . 13 𝑦 ∈ V
2726elixp 8471 . . . . . . . . . . . 12 (𝑦X𝑡𝐴 (𝐹𝑡) ↔ (𝑦 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡)))
2825, 27sylib 220 . . . . . . . . . . 11 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → (𝑦 Fn 𝐴 ∧ ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡)))
2928simprd 498 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡𝐴 (𝑦𝑡) ∈ (𝐹𝑡))
3029r19.21bi 3211 . . . . . . . . 9 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → (𝑦𝑡) ∈ (𝐹𝑡))
31 eqid 2824 . . . . . . . . . 10 (𝐹𝑡) = (𝐹𝑡)
3231pconncn 32475 . . . . . . . . 9 (((𝐹𝑡) ∈ PConn ∧ (𝑥𝑡) ∈ (𝐹𝑡) ∧ (𝑦𝑡) ∈ (𝐹𝑡)) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
3312, 23, 30, 32syl3anc 1367 . . . . . . . 8 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → ∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)))
34 df-rex 3147 . . . . . . . 8 (∃𝑓 ∈ (II Cn (𝐹𝑡))((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)) ↔ ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3533, 34sylib 220 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡𝐴) → ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3610, 35syldan 593 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ 𝑡 ∈ ( I ‘𝐴)) → ∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
3736ralrimiva 3185 . . . . 5 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∀𝑡 ∈ ( I ‘𝐴)∃𝑓(𝑓 ∈ (II Cn (𝐹𝑡)) ∧ ((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡))))
38 fvex 6686 . . . . . 6 ( I ‘𝐴) ∈ V
39 eleq1 2903 . . . . . . 7 (𝑓 = (𝑔𝑡) → (𝑓 ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑡) ∈ (II Cn (𝐹𝑡))))
40 fveq1 6672 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘0) = ((𝑔𝑡)‘0))
4140eqeq1d 2826 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘0) = (𝑥𝑡) ↔ ((𝑔𝑡)‘0) = (𝑥𝑡)))
42 fveq1 6672 . . . . . . . . 9 (𝑓 = (𝑔𝑡) → (𝑓‘1) = ((𝑔𝑡)‘1))
4342eqeq1d 2826 . . . . . . . 8 (𝑓 = (𝑔𝑡) → ((𝑓‘1) = (𝑦𝑡) ↔ ((𝑔𝑡)‘1) = (𝑦𝑡)))
4441, 43anbi12d 632 . . . . . . 7 (𝑓 = (𝑔𝑡) → (((𝑓‘0) = (𝑥𝑡) ∧ (𝑓‘1) = (𝑦𝑡)) ↔ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))
4539, 44anbi12d 632 . . . . . 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 9911 . . . . 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⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∃𝑔(𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)))))
48 iitopon 23490 . . . . . . 7 II ∈ (TopOn‘(0[,]1))
4948a1i 11 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 773 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐴𝑉)
5111adantr 483 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) ∧ (𝑔 Fn ( I ‘𝐴) ∧ ∀𝑡 ∈ ( I ‘𝐴)((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))))) → 𝐹:𝐴⟶PConn)
5251, 4syl 17 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 483 . . . . . . . . . . . . 13 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 2901 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 480 . . . . . . . . . . 11 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 771 . . . . . . . . . . . 12 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 6673 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (𝑔𝑡) = (𝑔𝑖))
58 fveq2 6673 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝐹𝑡) = (𝐹𝑖))
5958oveq2d 7175 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (II Cn (𝐹𝑡)) = (II Cn (𝐹𝑖)))
6057, 59eleq12d 2910 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ↔ (𝑔𝑖) ∈ (II Cn (𝐹𝑖))))
6157fveq1d 6675 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘0) = ((𝑔𝑖)‘0))
62 fveq2 6673 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑥𝑡) = (𝑥𝑖))
6361, 62eqeq12d 2840 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘0) = (𝑥𝑡) ↔ ((𝑔𝑖)‘0) = (𝑥𝑖)))
6457fveq1d 6675 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → ((𝑔𝑡)‘1) = ((𝑔𝑖)‘1))
65 fveq2 6673 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑖 → (𝑦𝑡) = (𝑦𝑖))
6664, 65eqeq12d 2840 . . . . . . . . . . . . . . 15 (𝑡 = 𝑖 → (((𝑔𝑡)‘1) = (𝑦𝑡) ↔ ((𝑔𝑖)‘1) = (𝑦𝑖)))
6763, 66anbi12d 632 . . . . . . . . . . . . . 14 (𝑡 = 𝑖 → ((((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡)) ↔ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖))))
6860, 67anbi12d 632 . . . . . . . . . . . . 13 (𝑡 = 𝑖 → (((𝑔𝑡) ∈ (II Cn (𝐹𝑡)) ∧ (((𝑔𝑡)‘0) = (𝑥𝑡) ∧ ((𝑔𝑡)‘1) = (𝑦𝑡))) ↔ ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) ∧ (((𝑔𝑖)‘0) = (𝑥𝑖) ∧ ((𝑔𝑖)‘1) = (𝑦𝑖)))))
6968rspccva 3625 . . . . . . . . . . . 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 582 . . . . . . . . . . 11 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 593 . . . . . . . . . 10 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 497 . . . . . . . . 9 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 23492 . . . . . . . . . 10 (0[,]1) = II
74 eqid 2824 . . . . . . . . . 10 (𝐹𝑖) = (𝐹𝑖)
7573, 74cnf 21857 . . . . . . . . 9 ((𝑔𝑖) ∈ (II Cn (𝐹𝑖)) → (𝑔𝑖):(0[,]1)⟶ (𝐹𝑖))
7672, 75syl 17 . . . . . . . 8 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 6736 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 2917 . . . . . 6 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 22238 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 498 . . . . . . . 8 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 497 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 5164 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 12858 . . . . . . 7 0 ∈ (0[,]1)
84 mptexg 6987 . . . . . . . 8 (𝐴𝑉 → (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V)
8550, 84syl 17 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 6673 . . . . . . . . 9 (𝑧 = 0 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘0))
8786mpteq2dv 5165 . . . . . . . 8 (𝑧 = 0 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
88 eqid 2824 . . . . . . . 8 (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))
8987, 88fvmptg 6769 . . . . . . 7 ((0 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘0)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = (𝑖𝐴 ↦ ((𝑔𝑖)‘0)))
9083, 85, 89sylancr 589 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 497 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑥 Fn 𝐴)
9291adantr 483 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 6727 . . . . . . 7 (𝑥 Fn 𝐴𝑥 = (𝑖𝐴 ↦ (𝑥𝑖)))
9492, 93sylib 220 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 2869 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 498 . . . . . . 7 (((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 5164 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 12859 . . . . . . 7 1 ∈ (0[,]1)
99 mptexg 6987 . . . . . . . 8 (𝐴𝑉 → (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V)
10050, 99syl 17 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 6673 . . . . . . . . 9 (𝑧 = 1 → ((𝑔𝑖)‘𝑧) = ((𝑔𝑖)‘1))
102101mpteq2dv 5165 . . . . . . . 8 (𝑧 = 1 → (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
103102, 88fvmptg 6769 . . . . . . 7 ((1 ∈ (0[,]1) ∧ (𝑖𝐴 ↦ ((𝑔𝑖)‘1)) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = (𝑖𝐴 ↦ ((𝑔𝑖)‘1)))
10498, 100, 103sylancr 589 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 497 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → 𝑦 Fn 𝐴)
106105adantr 483 . . . . . . 7 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 6727 . . . . . . 7 (𝑦 Fn 𝐴𝑦 = (𝑖𝐴 ↦ (𝑦𝑖)))
108106, 107sylib 220 . . . . . 6 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 2869 . . . . 5 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 6672 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0))
111110eqeq1d 2826 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘0) = 𝑥 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘0) = 𝑥))
112 fveq1 6672 . . . . . . . 8 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1))
113112eqeq1d 2826 . . . . . . 7 (𝑓 = (𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧))) → ((𝑓‘1) = 𝑦 ↔ ((𝑧 ∈ (0[,]1) ↦ (𝑖𝐴 ↦ ((𝑔𝑖)‘𝑧)))‘1) = 𝑦))
114111, 113anbi12d 632 . . . . . 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 3626 . . . . 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 834 . . . 4 ((((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (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 1935 . . 3 (((𝐴𝑉𝐹:𝐴⟶PConn) ∧ (𝑥 (∏t𝐹) ∧ 𝑦 (∏t𝐹))) → ∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
118117ralrimivva 3194 . 2 ((𝐴𝑉𝐹:𝐴⟶PConn) → ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
119 eqid 2824 . . 3 (∏t𝐹) = (∏t𝐹)
120119ispconn 32474 . 2 ((∏t𝐹) ∈ PConn ↔ ((∏t𝐹) ∈ Top ∧ ∀𝑥 (∏t𝐹)∀𝑦 (∏t𝐹)∃𝑓 ∈ (II Cn (∏t𝐹))((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
1216, 118, 120sylanbrc 585 1 ((𝐴𝑉𝐹:𝐴⟶PConn) → (∏t𝐹) ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wex 1779  wcel 2113  wral 3141  wrex 3142  Vcvv 3497  wss 3939   cuni 4841  cmpt 5149   I cid 5462   Fn wfn 6353  wf 6354  cfv 6358  (class class class)co 7159  Xcixp 8464  0cc0 10540  1c1 10541  [,]cicc 12744  tcpt 16715  Topctop 21504  TopOnctopon 21521   Cn ccn 21835  IIcii 23486  PConncpconn 32470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-reg 9059  ax-inf2 9107  ax-ac2 9888  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-er 8292  df-map 8411  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fi 8878  df-sup 8909  df-inf 8910  df-r1 9196  df-rank 9197  df-card 9371  df-ac 9545  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-icc 12748  df-seq 13373  df-exp 13433  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-topgen 16720  df-pt 16721  df-psmet 20540  df-xmet 20541  df-met 20542  df-bl 20543  df-mopn 20544  df-top 21505  df-topon 21522  df-bases 21557  df-cn 21838  df-cnp 21839  df-ii 23488  df-pconn 32472
This theorem is referenced by: (None)
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