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Theorem cnmpt2t 12501
 Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt21.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
cnmpt2t.b (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
Assertion
Ref Expression
cnmpt2t (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑀,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2t
Dummy variables 𝑣 𝑢 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5429 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩))
2 df-ov 5785 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐴)‘⟨𝑢, 𝑣⟩)
31, 2eqtr4di 2191 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣))
4 fveq2 5429 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩))
5 df-ov 5785 . . . . . . 7 (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = ((𝑥𝑋, 𝑦𝑌𝐵)‘⟨𝑢, 𝑣⟩)
64, 5eqtr4di 2191 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧) = (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣))
73, 6opeq12d 3721 . . . . 5 (𝑧 = ⟨𝑢, 𝑣⟩ → ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩ = ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
87mpompt 5871 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩)
9 nfcv 2282 . . . . . . 7 𝑥𝑢
10 nfmpo1 5846 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐴)
11 nfcv 2282 . . . . . . 7 𝑥𝑣
129, 10, 11nfov 5809 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
13 nfmpo1 5846 . . . . . . 7 𝑥(𝑥𝑋, 𝑦𝑌𝐵)
149, 13, 11nfov 5809 . . . . . 6 𝑥(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
1512, 14nfop 3729 . . . . 5 𝑥⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
16 nfcv 2282 . . . . . . 7 𝑦𝑢
17 nfmpo2 5847 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐴)
18 nfcv 2282 . . . . . . 7 𝑦𝑣
1916, 17, 18nfov 5809 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣)
20 nfmpo2 5847 . . . . . . 7 𝑦(𝑥𝑋, 𝑦𝑌𝐵)
2116, 20, 18nfov 5809 . . . . . 6 𝑦(𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)
2219, 21nfop 3729 . . . . 5 𝑦⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩
23 nfcv 2282 . . . . 5 𝑢⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
24 nfcv 2282 . . . . 5 𝑣⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩
25 oveq12 5791 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦))
26 oveq12 5791 . . . . . 6 ((𝑢 = 𝑥𝑣 = 𝑦) → (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣) = (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦))
2725, 26opeq12d 3721 . . . . 5 ((𝑢 = 𝑥𝑣 = 𝑦) → ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩ = ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
2815, 22, 23, 24, 27cbvmpo 5858 . . . 4 (𝑢𝑋, 𝑣𝑌 ↦ ⟨(𝑢(𝑥𝑋, 𝑦𝑌𝐴)𝑣), (𝑢(𝑥𝑋, 𝑦𝑌𝐵)𝑣)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
298, 28eqtri 2161 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩)
30 cnmpt21.j . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
31 cnmpt21.k . . . . 5 (𝜑𝐾 ∈ (TopOn‘𝑌))
32 txtopon 12470 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
3330, 31, 32syl2anc 409 . . . 4 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
34 toponuni 12221 . . . 4 ((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) → (𝑋 × 𝑌) = (𝐽 ×t 𝐾))
35 mpteq1 4020 . . . 4 ((𝑋 × 𝑌) = (𝐽 ×t 𝐾) → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩))
3633, 34, 353syl 17 . . 3 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩))
37 simp2 983 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑥𝑋)
38 simp3 984 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝑦𝑌)
39 cnmpt21.a . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
40 cntop2 12410 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
4139, 40syl 14 . . . . . . . . . . 11 (𝜑𝐿 ∈ Top)
42 toptopon2 12225 . . . . . . . . . . 11 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
4341, 42sylib 121 . . . . . . . . . 10 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
44 cnf2 12413 . . . . . . . . . 10 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿)) → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
4533, 43, 39, 44syl3anc 1217 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
46 eqid 2140 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐴) = (𝑥𝑋, 𝑦𝑌𝐴)
4746fmpo 6107 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 ↔ (𝑥𝑋, 𝑦𝑌𝐴):(𝑋 × 𝑌)⟶ 𝐿)
4845, 47sylibr 133 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐴 𝐿)
49 rsp2 2485 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐴 𝐿 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
5048, 49syl 14 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐴 𝐿))
51503impib 1180 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐴 𝐿)
5246ovmpt4g 5901 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐴 𝐿) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
5337, 38, 51, 52syl3anc 1217 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦) = 𝐴)
54 cnmpt2t.b . . . . . . . . . . . 12 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀))
55 cntop2 12410 . . . . . . . . . . . 12 ((𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀) → 𝑀 ∈ Top)
5654, 55syl 14 . . . . . . . . . . 11 (𝜑𝑀 ∈ Top)
57 toptopon2 12225 . . . . . . . . . . 11 (𝑀 ∈ Top ↔ 𝑀 ∈ (TopOn‘ 𝑀))
5856, 57sylib 121 . . . . . . . . . 10 (𝜑𝑀 ∈ (TopOn‘ 𝑀))
59 cnf2 12413 . . . . . . . . . 10 (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑀 ∈ (TopOn‘ 𝑀) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
6033, 58, 54, 59syl3anc 1217 . . . . . . . . 9 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
61 eqid 2140 . . . . . . . . . 10 (𝑥𝑋, 𝑦𝑌𝐵) = (𝑥𝑋, 𝑦𝑌𝐵)
6261fmpo 6107 . . . . . . . . 9 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 ↔ (𝑥𝑋, 𝑦𝑌𝐵):(𝑋 × 𝑌)⟶ 𝑀)
6360, 62sylibr 133 . . . . . . . 8 (𝜑 → ∀𝑥𝑋𝑦𝑌 𝐵 𝑀)
64 rsp2 2485 . . . . . . . 8 (∀𝑥𝑋𝑦𝑌 𝐵 𝑀 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
6563, 64syl 14 . . . . . . 7 (𝜑 → ((𝑥𝑋𝑦𝑌) → 𝐵 𝑀))
66653impib 1180 . . . . . 6 ((𝜑𝑥𝑋𝑦𝑌) → 𝐵 𝑀)
6761ovmpt4g 5901 . . . . . 6 ((𝑥𝑋𝑦𝑌𝐵 𝑀) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
6837, 38, 66, 67syl3anc 1217 . . . . 5 ((𝜑𝑥𝑋𝑦𝑌) → (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦) = 𝐵)
6953, 68opeq12d 3721 . . . 4 ((𝜑𝑥𝑋𝑦𝑌) → ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩ = ⟨𝐴, 𝐵⟩)
7069mpoeq3dva 5843 . . 3 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨(𝑥(𝑥𝑋, 𝑦𝑌𝐴)𝑦), (𝑥(𝑥𝑋, 𝑦𝑌𝐵)𝑦)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
7129, 36, 703eqtr3a 2197 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩))
72 eqid 2140 . . . 4 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
73 eqid 2140 . . . 4 (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) = (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩)
7472, 73txcnmpt 12481 . . 3 (((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ∧ (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐽 ×t 𝐾) Cn 𝑀)) → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
7539, 54, 74syl2anc 409 . 2 (𝜑 → (𝑧 (𝐽 ×t 𝐾) ↦ ⟨((𝑥𝑋, 𝑦𝑌𝐴)‘𝑧), ((𝑥𝑋, 𝑦𝑌𝐵)‘𝑧)⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
7671, 75eqeltrrd 2218 1 (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝐴, 𝐵⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐿 ×t 𝑀)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ⟨cop 3535  ∪ cuni 3744   ↦ cmpt 3997   × cxp 4545  ⟶wf 5127  ‘cfv 5131  (class class class)co 5782   ∈ cmpo 5784  Topctop 12203  TopOnctopon 12216   Cn ccn 12393   ×t ctx 12460 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-map 6552  df-topgen 12180  df-top 12204  df-topon 12217  df-bases 12249  df-cn 12396  df-tx 12461 This theorem is referenced by:  cnmpt22  12502  txhmeo  12527  txswaphmeo  12529
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