MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt2k Structured version   Visualization version   GIF version

Theorem cnmpt2k 23039
Description: The currying of a two-argument function is continuous. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypotheses
Ref Expression
cnmpt2k.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt2k.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt2k.a (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Assertion
Ref Expression
cnmpt2k (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2k
Dummy variables 𝑤 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2907 . . . . 5 𝑥𝑌
2 nfcv 2907 . . . . . 6 𝑥𝑣
3 nfmpo2 7438 . . . . . 6 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
4 nfcv 2907 . . . . . 6 𝑥𝑤
52, 3, 4nfov 7387 . . . . 5 𝑥(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
61, 5nfmpt 5212 . . . 4 𝑥(𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
7 nfcv 2907 . . . 4 𝑤(𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
8 nfcv 2907 . . . . . . 7 𝑦𝑣
9 nfmpo1 7437 . . . . . . 7 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
10 nfcv 2907 . . . . . . 7 𝑦𝑤
118, 9, 10nfov 7387 . . . . . 6 𝑦(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
12 nfcv 2907 . . . . . 6 𝑣(𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
13 oveq1 7364 . . . . . 6 (𝑣 = 𝑦 → (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
1411, 12, 13cbvmpt 5216 . . . . 5 (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
15 oveq2 7365 . . . . . 6 (𝑤 = 𝑥 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
1615mpteq2dv 5207 . . . . 5 (𝑤 = 𝑥 → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
1714, 16eqtrid 2788 . . . 4 (𝑤 = 𝑥 → (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
186, 7, 17cbvmpt 5216 . . 3 (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
19 simpr 485 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
20 simplr 767 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
21 cnmpt2k.k . . . . . . . . . . . 12 (𝜑𝐾 ∈ (TopOn‘𝑌))
22 cnmpt2k.j . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝑋))
23 txtopon 22942 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
2421, 22, 23syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
25 cnmpt2k.a . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
26 cntop2 22592 . . . . . . . . . . . . 13 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
28 toptopon2 22267 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
2927, 28sylib 217 . . . . . . . . . . 11 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
3022, 21, 25cnmptcom 23029 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
31 cnf2 22600 . . . . . . . . . . 11 (((𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿)) → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3224, 29, 30, 31syl3anc 1371 . . . . . . . . . 10 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
33 eqid 2736 . . . . . . . . . . 11 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
3433fmpo 8000 . . . . . . . . . 10 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3532, 34sylibr 233 . . . . . . . . 9 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
3635r19.21bi 3234 . . . . . . . 8 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴 𝐿)
3736r19.21bi 3234 . . . . . . 7 (((𝜑𝑦𝑌) ∧ 𝑥𝑋) → 𝐴 𝐿)
3837an32s 650 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 𝐿)
3933ovmpt4g 7502 . . . . . 6 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4019, 20, 38, 39syl3anc 1371 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4140mpteq2dva 5205 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) = (𝑦𝑌𝐴))
4241mpteq2dva 5205 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
4318, 42eqtrid 2788 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
44 eqid 2736 . . . . 5 (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩))
4544xkoinjcn 23038 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
4622, 21, 45syl2anc 584 . . 3 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ↑ko 𝐾)))
4732feqmptd 6910 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)))
4847, 30eqeltrrd 2839 . . 3 (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
49 fveq2 6842 . . . 4 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩))
50 df-ov 7360 . . . 4 (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩)
5149, 50eqtr4di 2794 . . 3 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
5222, 21, 24, 46, 48, 51cnmptk1 23032 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) ∈ (𝐽 Cn (𝐿ko 𝐾)))
5343, 52eqeltrrd 2839 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3064  cop 4592   cuni 4865  cmpt 5188   × cxp 5631  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  Topctop 22242  TopOnctopon 22259   Cn ccn 22575   ×t ctx 22911  ko cxko 22912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-cnp 22579  df-cmp 22738  df-tx 22913  df-xko 22914
This theorem is referenced by:  xkocnv  23165  xkohmeo  23166
  Copyright terms: Public domain W3C validator