Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  txpcon Structured version   Visualization version   GIF version

Theorem txpcon 30468
 Description: The topological product of two path-connected spaces is path-connected. (Contributed by Mario Carneiro, 12-Feb-2015.)
Assertion
Ref Expression
txpcon ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (𝑅 ×t 𝑆) ∈ PCon)

Proof of Theorem txpcon
Dummy variables 𝑓 𝑥 𝑦 𝑔 𝑡 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pcontop 30461 . . 3 (𝑅 ∈ PCon → 𝑅 ∈ Top)
2 pcontop 30461 . . 3 (𝑆 ∈ PCon → 𝑆 ∈ Top)
3 txtop 21182 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 493 . 2 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (𝑅 ×t 𝑆) ∈ Top)
5 an6 1400 . . . . . . . . . 10 (((𝑅 ∈ PCon ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PCon ∧ 𝑦 𝑆𝑤 𝑆)) ↔ ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)))
6 eqid 2610 . . . . . . . . . . . 12 𝑅 = 𝑅
76pconcn 30460 . . . . . . . . . . 11 ((𝑅 ∈ PCon ∧ 𝑥 𝑅𝑧 𝑅) → ∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧))
8 eqid 2610 . . . . . . . . . . . 12 𝑆 = 𝑆
98pconcn 30460 . . . . . . . . . . 11 ((𝑆 ∈ PCon ∧ 𝑦 𝑆𝑤 𝑆) → ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤))
107, 9anim12i 588 . . . . . . . . . 10 (((𝑅 ∈ PCon ∧ 𝑥 𝑅𝑧 𝑅) ∧ (𝑆 ∈ PCon ∧ 𝑦 𝑆𝑤 𝑆)) → (∃𝑔 ∈ (II Cn 𝑅)((𝑔‘0) = 𝑥 ∧ (𝑔‘1) = 𝑧) ∧ ∃ ∈ (II Cn 𝑆)((‘0) = 𝑦 ∧ (‘1) = 𝑤)))
115, 10sylbir 224 . . . . . . . . 9 (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 3086 . . . . . . . . 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 223 . . . . . . . 8 (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 22492 . . . . . . . . . . . . 13 (0[,]1) = II
15 eqid 2610 . . . . . . . . . . . . 13 (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)
1614, 15txcnmpt 21237 . . . . . . . . . . . 12 ((𝑔 ∈ (II Cn 𝑅) ∧ ∈ (II Cn 𝑆)) → (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) ∈ (II Cn (𝑅 ×t 𝑆)))
1716ad2antrl 760 . . . . . . . . . . 11 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 12161 . . . . . . . . . . . . 13 0 ∈ (0[,]1)
19 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑔𝑡) = (𝑔‘0))
20 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑡 = 0 → (𝑡) = (‘0))
2119, 20opeq12d 4348 . . . . . . . . . . . . . 14 (𝑡 = 0 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘0), (‘0)⟩)
22 opex 4859 . . . . . . . . . . . . . 14 ⟨(𝑔‘0), (‘0)⟩ ∈ V
2321, 15, 22fvmpt 6191 . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 474 . . . . . . . . . . . . 13 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 801 . . . . . . . . . . . . . 14 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 474 . . . . . . . . . . . . 13 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 4348 . . . . . . . . . . . 12 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 2656 . . . . . . . . . . 11 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 12162 . . . . . . . . . . . . 13 1 ∈ (0[,]1)
32 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑔𝑡) = (𝑔‘1))
33 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑡 = 1 → (𝑡) = (‘1))
3432, 33opeq12d 4348 . . . . . . . . . . . . . 14 (𝑡 = 1 → ⟨(𝑔𝑡), (𝑡)⟩ = ⟨(𝑔‘1), (‘1)⟩)
35 opex 4859 . . . . . . . . . . . . . 14 ⟨(𝑔‘1), (‘1)⟩ ∈ V
3634, 15, 35fvmpt 6191 . . . . . . . . . . . . 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 478 . . . . . . . . . . . . 13 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 478 . . . . . . . . . . . . 13 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 4348 . . . . . . . . . . . 12 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 2656 . . . . . . . . . . 11 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 6102 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘0) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0))
4342eqeq1d 2612 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘0) = ⟨𝑥, 𝑦⟩))
44 fveq1 6102 . . . . . . . . . . . . . 14 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → (𝑓‘1) = ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1))
4544eqeq1d 2612 . . . . . . . . . . . . 13 (𝑓 = (𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩) → ((𝑓‘1) = ⟨𝑧, 𝑤⟩ ↔ ((𝑡 ∈ (0[,]1) ↦ ⟨(𝑔𝑡), (𝑡)⟩)‘1) = ⟨𝑧, 𝑤⟩))
4643, 45anbi12d 743 . . . . . . . . . . . 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 3282 . . . . . . . . . . 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 1316 . . . . . . . . . 10 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 641 . . . . . . . . 9 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 3020 . . . . . . . 8 (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (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 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (𝑥 𝑅𝑦 𝑆) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
52513expa 1257 . . . . . 6 ((((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (𝑥 𝑅𝑦 𝑆)) ∧ (𝑧 𝑅𝑤 𝑆)) → ∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5352ralrimivva 2954 . . . . 5 (((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) ∧ (𝑥 𝑅𝑦 𝑆)) → ∀𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5453ralrimivva 2954 . . . 4 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → ∀𝑥 𝑅𝑦 𝑆𝑧 𝑅𝑤 𝑆𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
55 eqeq2 2621 . . . . . . . . 9 (𝑣 = ⟨𝑧, 𝑤⟩ → ((𝑓‘1) = 𝑣 ↔ (𝑓‘1) = ⟨𝑧, 𝑤⟩))
5655anbi2d 736 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣) ↔ ((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
5756rexbidv 3034 . . . . . . 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 5185 . . . . . 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 2621 . . . . . . . . 9 (𝑢 = ⟨𝑥, 𝑦⟩ → ((𝑓‘0) = 𝑢 ↔ (𝑓‘0) = ⟨𝑥, 𝑦⟩))
6059anbi1d 737 . . . . . . . 8 (𝑢 = ⟨𝑥, 𝑦⟩ → (((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩) ↔ ((𝑓‘0) = ⟨𝑥, 𝑦⟩ ∧ (𝑓‘1) = ⟨𝑧, 𝑤⟩)))
6160rexbidv 3034 . . . . . . 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 2972 . . . . . 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 271 . . . . 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 5185 . . . 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 223 . . 3 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → ∀𝑢 ∈ ( 𝑅 × 𝑆)∀𝑣 ∈ ( 𝑅 × 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
666, 8txuni 21205 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
671, 2, 66syl2an 493 . . . 4 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
6867raleqdv 3121 . . . 4 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (∀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 3129 . . 3 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (∀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 221 . 2 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → ∀𝑢 (𝑅 ×t 𝑆)∀𝑣 (𝑅 ×t 𝑆)∃𝑓 ∈ (II Cn (𝑅 ×t 𝑆))((𝑓‘0) = 𝑢 ∧ (𝑓‘1) = 𝑣))
71 eqid 2610 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
7271ispcon 30459 . 2 ((𝑅 ×t 𝑆) ∈ PCon ↔ ((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 695 1 ((𝑅 ∈ PCon ∧ 𝑆 ∈ PCon) → (𝑅 ×t 𝑆) ∈ PCon)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ⟨cop 4131  ∪ cuni 4372   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  [,]cicc 12049  Topctop 20517   Cn ccn 20838   ×t ctx 21173  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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  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-iun 4457  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-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-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-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  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-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-tx 21175  df-ii 22488  df-pcon 30457 This theorem is referenced by:  txscon  30477
 Copyright terms: Public domain W3C validator