![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uncfcl | Structured version Visualization version GIF version |
Description: The uncurry operation takes a functor 𝐹:𝐶⟶(𝐷⟶𝐸) to a functor uncurryF (𝐹):𝐶 × 𝐷⟶𝐸. (Contributed by Mario Carneiro, 13-Jan-2017.) |
Ref | Expression |
---|---|
uncfval.g | ⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) |
uncfval.c | ⊢ (𝜑 → 𝐷 ∈ Cat) |
uncfval.d | ⊢ (𝜑 → 𝐸 ∈ Cat) |
uncfval.f | ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) |
Ref | Expression |
---|---|
uncfcl | ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncfval.g | . . 3 ⊢ 𝐹 = (⟨“𝐶𝐷𝐸”⟩ uncurryF 𝐺) | |
2 | uncfval.c | . . 3 ⊢ (𝜑 → 𝐷 ∈ Cat) | |
3 | uncfval.d | . . 3 ⊢ (𝜑 → 𝐸 ∈ Cat) | |
4 | uncfval.f | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸))) | |
5 | 1, 2, 3, 4 | uncfval 18226 | . 2 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))) |
6 | eqid 2728 | . . . 4 ⊢ ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) | |
7 | eqid 2728 | . . . 4 ⊢ ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷) | |
8 | eqid 2728 | . . . . . 6 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
9 | funcrcl 17849 | . . . . . . . 8 ⊢ (𝐺 ∈ (𝐶 Func (𝐷 FuncCat 𝐸)) → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat)) | |
10 | 4, 9 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ (𝐷 FuncCat 𝐸) ∈ Cat)) |
11 | 10 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) |
12 | eqid 2728 | . . . . . 6 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
13 | 8, 11, 2, 12 | 1stfcl 18188 | . . . . 5 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) |
14 | 13, 4 | cofucl 17874 | . . . 4 ⊢ (𝜑 → (𝐺 ∘func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸))) |
15 | eqid 2728 | . . . . 5 ⊢ (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷) | |
16 | 8, 11, 2, 15 | 2ndfcl 18189 | . . . 4 ⊢ (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) |
17 | 6, 7, 14, 16 | prfcl 18194 | . . 3 ⊢ (𝜑 → ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷))) |
18 | eqid 2728 | . . . 4 ⊢ (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸) | |
19 | eqid 2728 | . . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸) | |
20 | 18, 19, 2, 3 | evlfcl 18214 | . . 3 ⊢ (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸)) |
21 | 17, 20 | cofucl 17874 | . 2 ⊢ (𝜑 → ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))) ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
22 | 5, 21 | eqeltrd 2829 | 1 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 (class class class)co 7420 ⟨“cs3 14826 Catccat 17644 Func cfunc 17840 ∘func ccofu 17842 FuncCat cfuc 17932 ×c cxpc 18159 1stF c1stf 18160 2ndF c2ndf 18161 ⟨,⟩F cprf 18162 evalF cevlf 18201 uncurryF cuncf 18203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-hash 14323 df-word 14498 df-concat 14554 df-s1 14579 df-s2 14832 df-s3 14833 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-hom 17257 df-cco 17258 df-cat 17648 df-cid 17649 df-func 17844 df-cofu 17846 df-nat 17933 df-fuc 17934 df-xpc 18163 df-1stf 18164 df-2ndf 18165 df-prf 18166 df-evlf 18205 df-uncf 18207 |
This theorem is referenced by: curfuncf 18230 uncfcurf 18231 |
Copyright terms: Public domain | W3C validator |