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

Theorem cnmpt2k 21414
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 (𝐿 ^ko 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 2761 . . . . 5 𝑥𝑌
2 nfcv 2761 . . . . . 6 𝑥𝑣
3 nfmpt22 6683 . . . . . 6 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
4 nfcv 2761 . . . . . 6 𝑥𝑤
52, 3, 4nfov 6636 . . . . 5 𝑥(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
61, 5nfmpt 4711 . . . 4 𝑥(𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
7 nfcv 2761 . . . 4 𝑤(𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
8 nfcv 2761 . . . . . . 7 𝑦𝑣
9 nfmpt21 6682 . . . . . . 7 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
10 nfcv 2761 . . . . . . 7 𝑦𝑤
118, 9, 10nfov 6636 . . . . . 6 𝑦(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
12 nfcv 2761 . . . . . 6 𝑣(𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
13 oveq1 6617 . . . . . 6 (𝑣 = 𝑦 → (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
1411, 12, 13cbvmpt 4714 . . . . 5 (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
15 oveq2 6618 . . . . . 6 (𝑤 = 𝑥 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
1615mpteq2dv 4710 . . . . 5 (𝑤 = 𝑥 → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
1714, 16syl5eq 2667 . . . 4 (𝑤 = 𝑥 → (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
186, 7, 17cbvmpt 4714 . . 3 (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
19 simpr 477 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
20 simplr 791 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
21 cnmpt2k.k . . . . . . . . . . . 12 (𝜑𝐾 ∈ (TopOn‘𝑌))
22 cnmpt2k.j . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝑋))
23 txtopon 21317 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
2421, 22, 23syl2anc 692 . . . . . . . . . . 11 (𝜑 → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
25 cnmpt2k.a . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
26 cntop2 20968 . . . . . . . . . . . . 13 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
28 eqid 2621 . . . . . . . . . . . . 13 𝐿 = 𝐿
2928toptopon 20657 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
3027, 29sylib 208 . . . . . . . . . . 11 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
3122, 21, 25cnmptcom 21404 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
32 cnf2 20976 . . . . . . . . . . 11 (((𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)) ∧ 𝐿 ∈ (TopOn‘ 𝐿) ∧ (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿)) → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3324, 30, 31, 32syl3anc 1323 . . . . . . . . . 10 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
34 eqid 2621 . . . . . . . . . . 11 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
3534fmpt2 7189 . . . . . . . . . 10 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3633, 35sylibr 224 . . . . . . . . 9 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
3736r19.21bi 2927 . . . . . . . 8 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴 𝐿)
3837r19.21bi 2927 . . . . . . 7 (((𝜑𝑦𝑌) ∧ 𝑥𝑋) → 𝐴 𝐿)
3938an32s 845 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 𝐿)
4034ovmpt4g 6743 . . . . . 6 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4119, 20, 39, 40syl3anc 1323 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4241mpteq2dva 4709 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) = (𝑦𝑌𝐴))
4342mpteq2dva 4709 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
4418, 43syl5eq 2667 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
45 eqid 2621 . . . . 5 (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩))
4645xkoinjcn 21413 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
4722, 21, 46syl2anc 692 . . 3 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
4833feqmptd 6211 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)))
4948, 31eqeltrrd 2699 . . 3 (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
50 fveq2 6153 . . . 4 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩))
51 df-ov 6613 . . . 4 (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩)
5250, 51syl6eqr 2673 . . 3 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
5322, 21, 24, 47, 49, 52cnmptk1 21407 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
5444, 53eqeltrrd 2699 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  cop 4159   cuni 4407  cmpt 4678   × cxp 5077  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612  Topctop 20630  TopOnctopon 20647   Cn ccn 20951   ×t ctx 21286   ^ko cxko 21287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-fin 7911  df-fi 8269  df-rest 16015  df-topgen 16036  df-top 20631  df-topon 20648  df-bases 20674  df-cn 20954  df-cnp 20955  df-cmp 21113  df-tx 21288  df-xko 21289
This theorem is referenced by:  xkocnv  21540  xkohmeo  21541
  Copyright terms: Public domain W3C validator