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Theorem txpconn 32376
Description: The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
Assertion
Ref Expression
txpconn ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)

Proof of Theorem txpconn
Dummy variables 𝑓 𝑥 𝑦 𝑔 𝑡 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pconntop 32369 . . 3 (𝑅 ∈ PConn → 𝑅 ∈ Top)
2 pconntop 32369 . . 3 (𝑆 ∈ PConn → 𝑆 ∈ Top)
3 txtop 22105 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 595 . 2 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ Top)
5 an6 1436 . . . . . . . . . 10 (((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆)) ↔ ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)))
6 eqid 2818 . . . . . . . . . . . 12 𝑅 = 𝑅
76pconncn 32368 . . . . . . . . . . 11 ((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8 eqid 2818 . . . . . . . . . . . 12 𝑆 = 𝑆
98pconncn 32368 . . . . . . . . . . 11 ((𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆) → ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤))
107, 9anim12i 612 . . . . . . . . . 10 (((𝑅 ∈ PConn ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PConn ∧ 𝑦 𝑆𝑤 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
115, 10sylbir 236 . . . . . . . . 9 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
12 reeanv 3365 . . . . . . . . 9 (∃𝑔 ∈ (II Cn 𝑅)∃ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)) ↔ (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
1311, 12sylibr 235 . . . . . . . 8 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑔 ∈ (II Cn 𝑅)∃ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
14 iiuni 23416 . . . . . . . . . . . . 13 (0[,]1) = II
15 eqid 2818 . . . . . . . . . . . . 13 (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)
1614, 15txcnmpt 22160 . . . . . . . . . . . 12 ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)))
1716ad2antrl 724 . . . . . . . . . . 11 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)))
18 0elunit 12843 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
19 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑔𝑡) = (𝑔‘0))
20 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑡) = (‘0))
2119, 20opeq12d 4803 . . . . . . . . . . . . . 14 (𝑡 = 0 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘0), (‘0)⟩)
22 opex 5347 . . . . . . . . . . . . . 14 ⟨(𝑔‘0), (‘0)⟩ ∈ V
2321, 15, 22fvmpt 6761 . . . . . . . . . . . . 13 (0 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨(𝑔‘0), (‘0)⟩)
2418, 23ax-mp 5 . . . . . . . . . . . 12 ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨(𝑔‘0), (‘0)⟩
25 simprrl 777 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
2625simpld 495 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (𝑔‘0) = 𝑥)
27 simprrr 778 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((‘0) = 𝑦 ∧ (‘1) = 𝑤))
2827simpld 495 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (‘0) = 𝑦)
2926, 28opeq12d 4803 . . . . . . . . . . . 12 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ⟨(𝑔‘0), (‘0)⟩ = ⟨𝑥, 𝑦⟩)
3024, 29syl5eq 2865 . . . . . . . . . . 11 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩)
31 1elunit 12844 . . . . . . . . . . . . 13 1 ∈ (0[,]1)
32 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑔𝑡) = (𝑔‘1))
33 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑡) = (‘1))
3432, 33opeq12d 4803 . . . . . . . . . . . . . 14 (𝑡 = 1 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘1), (‘1)⟩)
35 opex 5347 . . . . . . . . . . . . . 14 ⟨(𝑔‘1), (‘1)⟩ ∈ V
3634, 15, 35fvmpt 6761 . . . . . . . . . . . . 13 (1 ∈ (0[,]1) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨(𝑔‘1), (‘1)⟩)
3731, 36ax-mp 5 . . . . . . . . . . . 12 ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨(𝑔‘1), (‘1)⟩
3825simprd 496 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (𝑔‘1) = 𝑧)
3927simprd 496 . . . . . . . . . . . . 13 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → (‘1) = 𝑤)
4038, 39opeq12d 4803 . . . . . . . . . . . 12 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ⟨(𝑔‘1), (‘1)⟩ = ⟨𝑧, 𝑤⟩)
4137, 40syl5eq 2865 . . . . . . . . . . 11 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)
42 fveq1 6662 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0))
4342eqeq1d 2820 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩))
44 fveq1 6662 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1))
4544eqeq1d 2820 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘1) = ⟨𝑧, 𝑤⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩))
4643, 45anbi12d 630 . . . . . . . . . . . 12 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)))
4746rspcev 3620 . . . . . . . . . . 11 (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)) ∧ (((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩ ∧ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
4817, 30, 41, 47syl12anc 832 . . . . . . . . . 10 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) ∧ (((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)))) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
4948expr 457 . . . . . . . . 9 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) ∧ (𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆))) → ((((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5049rexlimdvva 3291 . . . . . . . 8 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)∃ ∈ (II Cn 𝑆)(((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ((‘0) = 𝑦 ∧ (‘1) = 𝑤)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5113, 50mpd 15 . . . . . . 7 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
52513expa 1110 . . . . . 6 ((((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆)) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5352ralrimivva 3188 . . . . 5 (((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) ∧ (𝑥 𝑅𝑦 𝑆)) → ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5453ralrimivva 3188 . . . 4 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑥 𝑅𝑦 𝑆𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
55 eqeq2 2830 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5655anbi2d 628 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5756rexbidv 3294 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5857ralxp 5705 . . . . . 6 (∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
59 eqeq2 2830 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
6059anbi1d 629 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6160rexbidv 3294 . . . . . . 7 (𝑢 = ⟨𝑥, 𝑦⟩ → (∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
62612ralbidv 3196 . . . . . 6 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6358, 62syl5bb 284 . . . . 5 (𝑢 = ⟨𝑥, 𝑦⟩ → (∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6463ralxp 5705 . . . 4 (∀𝑢 ∈ ( 𝑅 × 𝑆)∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑥 𝑅𝑦 𝑆𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
6554, 64sylibr 235 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 ∈ ( 𝑅 × 𝑆)∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
666, 8txuni 22128 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
671, 2, 66syl2an 595 . . . 4 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6867raleqdv 3413 . . . 4 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
6967, 68raleqbidv 3399 . . 3 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (∀𝑢 ∈ ( 𝑅 × 𝑆)∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ∀𝑢 (𝑅 ×t 𝑆)∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
7065, 69mpbid 233 . 2 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → ∀𝑢 (𝑅 ×t 𝑆)∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
71 eqid 2818 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7271ispconn 32367 . 2 ((𝑅 ×t 𝑆) ∈ PConn ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑢 (𝑅 ×t 𝑆)∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣)))
734, 70, 72sylanbrc 583 1 ((𝑅 ∈ PConn ∧ 𝑆 ∈ PConn) → (𝑅 ×t 𝑆) ∈ PConn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cop 4563   cuni 4830  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  [,]cicc 12729  Topctop 21429   Cn ccn 21760   ×t ctx 22096  IIcii 23410  PConncpconn 32363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-icc 12733  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-tx 22098  df-ii 23412  df-pconn 32365
This theorem is referenced by:  txsconn  32385
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