coccom - Mathbox for Zhi Wang

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Theorem coccom 49946

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 17880	. . 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 17884	. 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 49506	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 49362	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49361	. . . . 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 18168	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
19	18	oveq2d 7374	. . 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 7375	. 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 18168	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
22	21	opeq2d 4835	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ = ⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩)
23	22, 18	oveq12d 7376	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ · ((1^st ‘𝐾)‘𝑍)) = (⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋))
24		eqid 2735	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
25	11, 13, 14, 15, 16, 17, 4, 7, 5, 24, 8, 9	diag12 18169	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
26		eqidd 2736	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) = (𝑅‘𝑌))
27	23, 25, 26	oveq123d 7379	. . 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 2735	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
29	4, 15, 12	funcf1 17792	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
30	29, 7	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ 𝐴)
31	11, 15, 1, 4, 17, 16, 7, 28, 2	coccl 49944	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐶)𝑋))
32	15, 28, 24, 13, 30, 6, 16, 31	catlid 17608	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋)(𝑅‘𝑌)) = (𝑅‘𝑌))
33	27, 32	eqtrd 2770	. 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 2779	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

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4585 ‘cfv 6491 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 Basecbs 17138 Hom chom 17190 compcco 17191 Idccid 17590 Nat cnat 17870 Δ_funccdiag 18137

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2183 ax-ext 2707 ax-rep 5223 ax-sep 5240 ax-nul 5250 ax-pow 5309 ax-pr 5376 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3349 df-reu 3350 df-rab 3399 df-v 3441 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4285 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-iun 4947 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6258 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6447 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8767 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-fz 13426 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-hom 17203 df-cco 17204 df-cat 17593 df-cid 17594 df-func 17784 df-nat 17872 df-xpc 18097 df-1stf 18098 df-curf 18139 df-diag 18141

This theorem is referenced by: (None)

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