coccom - Mathbox for Zhi Wang

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Theorem coccom 51172

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 18170	. . 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 18174	. 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 50665	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 50519	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 50518	. . . . 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 18458	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
19	18	oveq2d 7465	. . 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 7466	. 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 18458	. . . . . 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 7467	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ · ((1^st ‘𝐾)‘𝑍)) = (⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋))
24		eqid 2763	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
25	11, 13, 14, 15, 16, 17, 4, 7, 5, 24, 8, 9	diag12 18459	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
26		eqidd 2764	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) = (𝑅‘𝑌))
27	23, 25, 26	oveq123d 7470	. . 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 2763	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
29	4, 15, 12	funcf1 18082	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
30	29, 7	ffvelcdmd 7115	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ 𝐴)
31	11, 15, 1, 4, 17, 16, 7, 28, 2	coccl 51170	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐶)𝑋))
32	15, 28, 24, 13, 30, 6, 16, 31	catlid 17898	. . 3 ⊢ (𝜑 → (((Id‘𝐶)‘𝑋)(⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩ · 𝑋)(𝑅‘𝑌)) = (𝑅‘𝑌))
33	27, 32	eqtrd 2798	. 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 2807	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 2147 ⟨cop 4838 ‘cfv 6564 (class class class)co 7449 1^st c1st 8031 2^nd c2nd 8032 Basecbs 17428 Hom chom 17480 compcco 17481 Idccid 17880 Nat cnat 18160 Δ_funccdiag 18427

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-om 7910 df-1st 8033 df-2nd 8034 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-er 8753 df-map 8885 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-nn 12374 df-2 12443 df-3 12444 df-4 12445 df-5 12446 df-6 12447 df-7 12448 df-8 12449 df-9 12450 df-n0 12645 df-z 12732 df-dec 12853 df-uz 13004 df-fz 13678 df-struct 17366 df-slot 17401 df-ndx 17413 df-base 17429 df-hom 17493 df-cco 17494 df-cat 17883 df-cid 17884 df-func 18074 df-nat 18162 df-xpc 18387 df-1stf 18388 df-curf 18429 df-diag 18431

This theorem is used by: (None)

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