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

Theorem cnmpt2k 21869
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 2969 . . . . 5 𝑥𝑌
2 nfcv 2969 . . . . . 6 𝑥𝑣
3 nfmpt22 6988 . . . . . 6 𝑥(𝑦𝑌, 𝑥𝑋𝐴)
4 nfcv 2969 . . . . . 6 𝑥𝑤
52, 3, 4nfov 6940 . . . . 5 𝑥(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
61, 5nfmpt 4971 . . . 4 𝑥(𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
7 nfcv 2969 . . . 4 𝑤(𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
8 nfcv 2969 . . . . . . 7 𝑦𝑣
9 nfmpt21 6987 . . . . . . 7 𝑦(𝑦𝑌, 𝑥𝑋𝐴)
10 nfcv 2969 . . . . . . 7 𝑦𝑤
118, 9, 10nfov 6940 . . . . . 6 𝑦(𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
12 nfcv 2969 . . . . . 6 𝑣(𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)
13 oveq1 6917 . . . . . 6 (𝑣 = 𝑦 → (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
1411, 12, 13cbvmpt 4974 . . . . 5 (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
15 oveq2 6918 . . . . . 6 (𝑤 = 𝑥 → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))
1615mpteq2dv 4970 . . . . 5 (𝑤 = 𝑥 → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
1714, 16syl5eq 2873 . . . 4 (𝑤 = 𝑥 → (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤)) = (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
186, 7, 17cbvmpt 4974 . . 3 (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)))
19 simpr 479 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑦𝑌)
20 simplr 785 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝑥𝑋)
21 cnmpt2k.k . . . . . . . . . . . 12 (𝜑𝐾 ∈ (TopOn‘𝑌))
22 cnmpt2k.j . . . . . . . . . . . 12 (𝜑𝐽 ∈ (TopOn‘𝑋))
23 txtopon 21772 . . . . . . . . . . . 12 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
2421, 22, 23syl2anc 579 . . . . . . . . . . 11 (𝜑 → (𝐾 ×t 𝐽) ∈ (TopOn‘(𝑌 × 𝑋)))
25 cnmpt2k.a . . . . . . . . . . . . 13 (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
26 cntop2 21423 . . . . . . . . . . . . 13 ((𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐽 ×t 𝐾) Cn 𝐿) → 𝐿 ∈ Top)
2725, 26syl 17 . . . . . . . . . . . 12 (𝜑𝐿 ∈ Top)
28 eqid 2825 . . . . . . . . . . . . 13 𝐿 = 𝐿
2928toptopon 21099 . . . . . . . . . . . 12 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
3027, 29sylib 210 . . . . . . . . . . 11 (𝜑𝐿 ∈ (TopOn‘ 𝐿))
3122, 21, 25cnmptcom 21859 . . . . . . . . . . 11 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
32 cnf2 21431 . . . . . . . . . . 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 1494 . . . . . . . . . 10 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
34 eqid 2825 . . . . . . . . . . 11 (𝑦𝑌, 𝑥𝑋𝐴) = (𝑦𝑌, 𝑥𝑋𝐴)
3534fmpt2 7505 . . . . . . . . . 10 (∀𝑦𝑌𝑥𝑋 𝐴 𝐿 ↔ (𝑦𝑌, 𝑥𝑋𝐴):(𝑌 × 𝑋)⟶ 𝐿)
3633, 35sylibr 226 . . . . . . . . 9 (𝜑 → ∀𝑦𝑌𝑥𝑋 𝐴 𝐿)
3736r19.21bi 3141 . . . . . . . 8 ((𝜑𝑦𝑌) → ∀𝑥𝑋 𝐴 𝐿)
3837r19.21bi 3141 . . . . . . 7 (((𝜑𝑦𝑌) ∧ 𝑥𝑋) → 𝐴 𝐿)
3938an32s 642 . . . . . 6 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → 𝐴 𝐿)
4034ovmpt4g 7048 . . . . . 6 ((𝑦𝑌𝑥𝑋𝐴 𝐿) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4119, 20, 39, 40syl3anc 1494 . . . . 5 (((𝜑𝑥𝑋) ∧ 𝑦𝑌) → (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥) = 𝐴)
4241mpteq2dva 4969 . . . 4 ((𝜑𝑥𝑋) → (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥)) = (𝑦𝑌𝐴))
4342mpteq2dva 4969 . . 3 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑦(𝑦𝑌, 𝑥𝑋𝐴)𝑥))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
4418, 43syl5eq 2873 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) = (𝑥𝑋 ↦ (𝑦𝑌𝐴)))
45 eqid 2825 . . . . 5 (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) = (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩))
4645xkoinjcn 21868 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
4722, 21, 46syl2anc 579 . . 3 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ ⟨𝑣, 𝑤⟩)) ∈ (𝐽 Cn ((𝐾 ×t 𝐽) ^ko 𝐾)))
4833feqmptd 6500 . . . 4 (𝜑 → (𝑦𝑌, 𝑥𝑋𝐴) = (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)))
4948, 31eqeltrrd 2907 . . 3 (𝜑 → (𝑧 ∈ (𝑌 × 𝑋) ↦ ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧)) ∈ ((𝐾 ×t 𝐽) Cn 𝐿))
50 fveq2 6437 . . . 4 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩))
51 df-ov 6913 . . . 4 (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤) = ((𝑦𝑌, 𝑥𝑋𝐴)‘⟨𝑣, 𝑤⟩)
5250, 51syl6eqr 2879 . . 3 (𝑧 = ⟨𝑣, 𝑤⟩ → ((𝑦𝑌, 𝑥𝑋𝐴)‘𝑧) = (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))
5322, 21, 24, 47, 49, 52cnmptk1 21862 . 2 (𝜑 → (𝑤𝑋 ↦ (𝑣𝑌 ↦ (𝑣(𝑦𝑌, 𝑥𝑋𝐴)𝑤))) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
5444, 53eqeltrrd 2907 1 (𝜑 → (𝑥𝑋 ↦ (𝑦𝑌𝐴)) ∈ (𝐽 Cn (𝐿 ^ko 𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  cop 4405   cuni 4660  cmpt 4954   × cxp 5344  wf 6123  cfv 6127  (class class class)co 6910  cmpt2 6912  Topctop 21075  TopOnctopon 21092   Cn ccn 21406   ×t ctx 21741   ^ko cxko 21742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-fin 8232  df-fi 8592  df-rest 16443  df-topgen 16464  df-top 21076  df-topon 21093  df-bases 21128  df-cn 21409  df-cnp 21410  df-cmp 21568  df-tx 21743  df-xko 21744
This theorem is referenced by:  xkocnv  21995  xkohmeo  21996
  Copyright terms: Public domain W3C validator