fuclid - Metamath Proof Explorer

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

Theorem fuclid 17815

Description: Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypotheses**
Ref	Expression
fuclid.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fuclid.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
fuclid.x	⊢ ∙ = (comp‘𝑄)
fuclid.1	⊢ 1 = (Id‘𝐷)
fuclid.r	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))

**Assertion**
Ref	Expression
fuclid	⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = 𝑅)

Proof of Theorem fuclid Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2738	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
3		relfunc 17708	. . . . . . . 8 ⊢ Rel (𝐶 Func 𝐷)
4		fuclid.r	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐺))
5		fuclid.n	. . . . . . . . . . 11 ⊢ 𝑁 = (𝐶 Nat 𝐷)
6	5	natrcl 17797	. . . . . . . . . 10 ⊢ (𝑅 ∈ (𝐹𝑁𝐺) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
7	4, 6	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)))
8	7	simprd 497	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷))
9		1st2ndbr 7967	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
10	3, 8, 9	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐺)(𝐶 Func 𝐷)(2^nd ‘𝐺))
11	1, 2, 10	funcf1 17712	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷))
12		fvco3 6938	. . . . . 6 ⊢ (((1^st ‘𝐺):(Base‘𝐶)⟶(Base‘𝐷) ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐺))‘𝑥) = ( 1 ‘((1^st ‘𝐺)‘𝑥)))
13	11, 12	sylan 581	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ∘ (1^st ‘𝐺))‘𝑥) = ( 1 ‘((1^st ‘𝐺)‘𝑥)))
14	13	oveq1d 7367	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)) = (( 1 ‘((1^st ‘𝐺)‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)))
15		eqid 2738	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		fuclid.1	. . . . 5 ⊢ 1 = (Id‘𝐷)
17	7	simpld 496	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
18		funcrcl 17709	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
20	19	simprd 497	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
21	20	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
22		1st2ndbr 7967	. . . . . . . 8 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
23	3, 17, 22	sylancr 588	. . . . . . 7 ⊢ (𝜑 → (1^st ‘𝐹)(𝐶 Func 𝐷)(2^nd ‘𝐹))
24	1, 2, 23	funcf1 17712	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹):(Base‘𝐶)⟶(Base‘𝐷))
25	24	ffvelcdmda 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐹)‘𝑥) ∈ (Base‘𝐷))
26		eqid 2738	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
27	11	ffvelcdmda 7032	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((1^st ‘𝐺)‘𝑥) ∈ (Base‘𝐷))
28	5, 4	nat1st2nd 17798	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
29	28	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐺), (2^nd ‘𝐺)⟩))
30		simpr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
31	5, 29, 1, 15, 30	natcl 17800	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑅‘𝑥) ∈ (((1^st ‘𝐹)‘𝑥)(Hom ‘𝐷)((1^st ‘𝐺)‘𝑥)))
32	2, 15, 16, 21, 25, 26, 27, 31	catlid 17523	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → (( 1 ‘((1^st ‘𝐺)‘𝑥))(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)) = (𝑅‘𝑥))
33	14, 32	eqtrd 2778	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐶)) → ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥)) = (𝑅‘𝑥))
34	33	mpteq2dva 5204	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥))) = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
35		fuclid.q	. . 3 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
36		fuclid.x	. . 3 ⊢ ∙ = (comp‘𝑄)
37	35, 5, 16, 8	fucidcl 17814	. . 3 ⊢ (𝜑 → ( 1 ∘ (1^st ‘𝐺)) ∈ (𝐺𝑁𝐺))
38	35, 5, 1, 26, 36, 4, 37	fucco 17811	. 2 ⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = (𝑥 ∈ (Base‘𝐶) ↦ ((( 1 ∘ (1^st ‘𝐺))‘𝑥)(⟨((1^st ‘𝐹)‘𝑥), ((1^st ‘𝐺)‘𝑥)⟩(comp‘𝐷)((1^st ‘𝐺)‘𝑥))(𝑅‘𝑥))))
39	5, 28, 1	natfn 17801	. . 3 ⊢ (𝜑 → 𝑅 Fn (Base‘𝐶))
40		dffn5 6899	. . 3 ⊢ (𝑅 Fn (Base‘𝐶) ↔ 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
41	39, 40	sylib 217	. 2 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ (Base‘𝐶) ↦ (𝑅‘𝑥)))
42	34, 38, 41	3eqtr4d 2788	1 ⊢ (𝜑 → (( 1 ∘ (1^st ‘𝐺))(⟨𝐹, 𝐺⟩ ∙ 𝐺)𝑅) = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4591 class class class wbr 5104 ↦ cmpt 5187 ∘ ccom 5636 Rel wrel 5637 Fn wfn 6489 ⟶wf 6490 ‘cfv 6494 (class class class)co 7352 1^st c1st 7912 2^nd c2nd 7913 Basecbs 17043 Hom chom 17104 compcco 17105 Catccat 17504 Idccid 17505 Func cfunc 17700 Nat cnat 17788 FuncCat cfuc 17789

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7665 ax-cnex 11066 ax-resscn 11067 ax-1cn 11068 ax-icn 11069 ax-addcl 11070 ax-addrcl 11071 ax-mulcl 11072 ax-mulrcl 11073 ax-mulcom 11074 ax-addass 11075 ax-mulass 11076 ax-distr 11077 ax-i2m1 11078 ax-1ne0 11079 ax-1rid 11080 ax-rnegex 11081 ax-rrecex 11082 ax-cnre 11083 ax-pre-lttri 11084 ax-pre-lttrn 11085 ax-pre-ltadd 11086 ax-pre-mulgt0 11087

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7308 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7796 df-1st 7914 df-2nd 7915 df-frecs 8205 df-wrecs 8236 df-recs 8310 df-rdg 8349 df-1o 8405 df-er 8607 df-map 8726 df-ixp 8795 df-en 8843 df-dom 8844 df-sdom 8845 df-fin 8846 df-pnf 11150 df-mnf 11151 df-xr 11152 df-ltxr 11153 df-le 11154 df-sub 11346 df-neg 11347 df-nn 12113 df-2 12175 df-3 12176 df-4 12177 df-5 12178 df-6 12179 df-7 12180 df-8 12181 df-9 12182 df-n0 12373 df-z 12459 df-dec 12578 df-uz 12723 df-fz 13380 df-struct 16979 df-slot 17014 df-ndx 17026 df-base 17044 df-hom 17117 df-cco 17118 df-cat 17508 df-cid 17509 df-func 17704 df-nat 17790 df-fuc 17791

This theorem is referenced by: fuccatid 17818

W3C validator