coccom - Mathbox for Zhi Wang

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Theorem coccom 50483

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 18036	. . 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 18040	. 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 50043	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 49899	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49898	. . . . 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 18324	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
19	18	oveq2d 7439	. . 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 7440	. 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 18324	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
22	21	opeq2d 4850	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ = ⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩)
23	22, 18	oveq12d 7441	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ · ((1^st ‘𝐾)‘𝑍)) = (⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋))
24		eqid 2766	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
25	11, 13, 14, 15, 16, 17, 4, 7, 5, 24, 8, 9	diag12 18325	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
26		eqidd 2767	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) = (𝑅‘𝑌))
27	23, 25, 26	oveq123d 7444	. . 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 2766	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
29	4, 15, 12	funcf1 17948	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
30	29, 7	ffvelcdmd 7087	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ 𝐴)
31	11, 15, 1, 4, 17, 16, 7, 28, 2	coccl 50481	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐶)𝑋))
32	15, 28, 24, 13, 30, 6, 16, 31	catlid 17764	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋)(𝑅‘𝑌)) = (𝑅‘𝑌))
33	27, 32	eqtrd 2801	. 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 2810	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 1570 ∈ wcel 2146 ⟨cop 4600 ‘cfv 6543 (class class class)co 7423 1^st c1st 7993 2^nd c2nd 7994 Basecbs 17294 Hom chom 17346 compcco 17347 Idccid 17746 Nat cnat 18026 Δ_funccdiag 18293

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-fz 13554 df-struct 17232 df-slot 17267 df-ndx 17279 df-base 17295 df-hom 17359 df-cco 17360 df-cat 17749 df-cid 17750 df-func 17940 df-nat 18028 df-xpc 18253 df-1stf 18254 df-curf 18295 df-diag 18297

This theorem is used by: (None)

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