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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 18195 | . 2 ⊢ (𝜑 → 𝐹 = ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)))) |
6 | eqid 2724 | . . . 4 ⊢ ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) = ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) | |
7 | eqid 2724 | . . . 4 ⊢ ((𝐷 FuncCat 𝐸) ×c 𝐷) = ((𝐷 FuncCat 𝐸) ×c 𝐷) | |
8 | eqid 2724 | . . . . . 6 ⊢ (𝐶 ×c 𝐷) = (𝐶 ×c 𝐷) | |
9 | funcrcl 17818 | . . . . . . . 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 2724 | . . . . . 6 ⊢ (𝐶 1stF 𝐷) = (𝐶 1stF 𝐷) | |
13 | 8, 11, 2, 12 | 1stfcl 18157 | . . . . 5 ⊢ (𝜑 → (𝐶 1stF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐶)) |
14 | 13, 4 | cofucl 17843 | . . . 4 ⊢ (𝜑 → (𝐺 ∘func (𝐶 1stF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func (𝐷 FuncCat 𝐸))) |
15 | eqid 2724 | . . . . 5 ⊢ (𝐶 2ndF 𝐷) = (𝐶 2ndF 𝐷) | |
16 | 8, 11, 2, 15 | 2ndfcl 18158 | . . . 4 ⊢ (𝜑 → (𝐶 2ndF 𝐷) ∈ ((𝐶 ×c 𝐷) Func 𝐷)) |
17 | 6, 7, 14, 16 | prfcl 18163 | . . 3 ⊢ (𝜑 → ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷)) ∈ ((𝐶 ×c 𝐷) Func ((𝐷 FuncCat 𝐸) ×c 𝐷))) |
18 | eqid 2724 | . . . 4 ⊢ (𝐷 evalF 𝐸) = (𝐷 evalF 𝐸) | |
19 | eqid 2724 | . . . 4 ⊢ (𝐷 FuncCat 𝐸) = (𝐷 FuncCat 𝐸) | |
20 | 18, 19, 2, 3 | evlfcl 18183 | . . 3 ⊢ (𝜑 → (𝐷 evalF 𝐸) ∈ (((𝐷 FuncCat 𝐸) ×c 𝐷) Func 𝐸)) |
21 | 17, 20 | cofucl 17843 | . 2 ⊢ (𝜑 → ((𝐷 evalF 𝐸) ∘func ((𝐺 ∘func (𝐶 1stF 𝐷)) ⟨,⟩F (𝐶 2ndF 𝐷))) ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
22 | 5, 21 | eqeltrd 2825 | 1 ⊢ (𝜑 → 𝐹 ∈ ((𝐶 ×c 𝐷) Func 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 (class class class)co 7402 ⟨“cs3 14795 Catccat 17613 Func cfunc 17809 ∘func ccofu 17811 FuncCat cfuc 17901 ×c cxpc 18128 1stF c1stf 18129 2ndF c2ndf 18130 ⟨,⟩F cprf 18131 evalF cevlf 18170 uncurryF cuncf 18172 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-hash 14292 df-word 14467 df-concat 14523 df-s1 14548 df-s2 14801 df-s3 14802 df-struct 17085 df-slot 17120 df-ndx 17132 df-base 17150 df-hom 17226 df-cco 17227 df-cat 17617 df-cid 17618 df-func 17813 df-cofu 17815 df-nat 17902 df-fuc 17903 df-xpc 18132 df-1stf 18133 df-2ndf 18134 df-prf 18135 df-evlf 18174 df-uncf 18176 |
This theorem is referenced by: curfuncf 18199 uncfcurf 18200 |
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