coccom - Mathbox for Zhi Wang

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Theorem coccom 50941

Description: A co-cone to a diagram commutes with the diagram. (Contributed by Zhi Wang, 13-Nov-2025.)

**Hypotheses**
Ref	Expression
islmd.l	⊢ 𝐿 = (𝐶Δ_func𝐷)
islmd.a	⊢ 𝐴 = (Base‘𝐶)
islmd.n	⊢ 𝑁 = (𝐷 Nat 𝐶)
islmd.b	⊢ 𝐵 = (Base‘𝐷)
concl.k	⊢ 𝐾 = ((1^st ‘𝐿)‘𝑋)
concl.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
concl.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
concom.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
concom.m	⊢ (𝜑 → 𝑀 ∈ (𝑌𝐽𝑍))
concom.j	⊢ 𝐽 = (Hom ‘𝐷)
concom.o	⊢ · = (comp‘𝐶)
coccom.r	⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐾))

**Assertion**
Ref	Expression
coccom	⊢ (𝜑 → (𝑅‘𝑌) = ((𝑅‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩ · 𝑋)((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)))

**Proof of Theorem coccom**
Step	Hyp	Ref	Expression
1		islmd.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐶)
2		coccom.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝐹𝑁𝐾))
3	1, 2	nat1st2nd 18135	. . 3 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩𝑁⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩))
4		islmd.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
5		concom.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐷)
6		concom.o	. . 3 ⊢ · = (comp‘𝐶)
7		concl.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		concom.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
9		concom.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑌𝐽𝑍))
10	1, 3, 4, 5, 6, 7, 8, 9	nati 18139	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩ · ((1^st ‘𝐾)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)) = (((𝑌(2^nd ‘𝐾)𝑍)‘𝑀)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ · ((1^st ‘𝐾)‘𝑍))(𝑅‘𝑌)))
11		islmd.l	. . . . 5 ⊢ 𝐿 = (𝐶Δ_func𝐷)
12	1, 3	natrcl2 50434	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 50288	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 50287	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Cat)
15		islmd.a	. . . . 5 ⊢ 𝐴 = (Base‘𝐶)
16		concl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
17		concl.k	. . . . 5 ⊢ 𝐾 = ((1^st ‘𝐿)‘𝑋)
18	11, 13, 14, 15, 16, 17, 4, 8	diag11 18423	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
19	18	oveq2d 7437	. . 3 ⊢ (𝜑 → (⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩ · ((1^st ‘𝐾)‘𝑍)) = (⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩ · 𝑋))
20	19	oveqd 7438	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩ · ((1^st ‘𝐾)‘𝑍))((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)) = ((𝑅‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩ · 𝑋)((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)))
21	11, 13, 14, 15, 16, 17, 4, 7	diag11 18423	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
22	21	opeq2d 4846	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ = ⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩)
23	22, 18	oveq12d 7439	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ · ((1^st ‘𝐾)‘𝑍)) = (⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋))
24		eqid 2760	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
25	11, 13, 14, 15, 16, 17, 4, 7, 5, 24, 8, 9	diag12 18424	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
26		eqidd 2761	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) = (𝑅‘𝑌))
27	23, 25, 26	oveq123d 7442	. . 3 ⊢ (𝜑 → (((𝑌(2^nd ‘𝐾)𝑍)‘𝑀)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ · ((1^st ‘𝐾)‘𝑍))(𝑅‘𝑌)) = (((Id‘𝐶)‘𝑋)(⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋)(𝑅‘𝑌)))
28		eqid 2760	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
29	4, 15, 12	funcf1 18047	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
30	29, 7	ffvelcdmd 7086	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ 𝐴)
31	11, 15, 1, 4, 17, 16, 7, 28, 2	coccl 50939	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐶)𝑋))
32	15, 28, 24, 13, 30, 6, 16, 31	catlid 17863	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋)(𝑅‘𝑌)) = (𝑅‘𝑌))
33	27, 32	eqtrd 2795	. 2 ⊢ (𝜑 → (((𝑌(2^nd ‘𝐾)𝑍)‘𝑀)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ · ((1^st ‘𝐾)‘𝑍))(𝑅‘𝑌)) = (𝑅‘𝑌))
34	10, 20, 33	3eqtr3rd 2804	1 ⊢ (𝜑 → (𝑅‘𝑌) = ((𝑅‘𝑍)(⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐹)‘𝑍)⟩ · 𝑋)((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 Basecbs 17393 Hom chom 17445 compcco 17446 Idccid 17845 Nat cnat 18125 Δ_funccdiag 18392

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-map 8853 df-ixp 8930 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-7 12414 df-8 12415 df-9 12416 df-n0 12611 df-z 12698 df-dec 12819 df-uz 12970 df-fz 13644 df-struct 17331 df-slot 17366 df-ndx 17378 df-base 17394 df-hom 17458 df-cco 17459 df-cat 17848 df-cid 17849 df-func 18039 df-nat 18127 df-xpc 18352 df-1stf 18353 df-curf 18394 df-diag 18396

This theorem is used by: (None)

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