coccom - Mathbox for Zhi Wang

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Theorem coccom 50590

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 18043	. . 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 18047	. 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 50150	. . . . . 6 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 50006	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 50005	. . . . 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 18331	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
19	18	oveq2d 7429	. . 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 7430	. 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 18331	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
22	21	opeq2d 4840	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐹)‘𝑌), ((1^st ‘𝐾)‘𝑌)⟩ = ⟨((1^st ‘𝐹)‘𝑌), 𝑋⟩)
23	22, 18	oveq12d 7431	. . . 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 18332	. . . 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 7434	. . 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 17955	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
30	29, 7	ffvelcdmd 7078	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑌) ∈ 𝐴)
31	11, 15, 1, 4, 17, 16, 7, 28, 2	coccl 50588	. . . 4 ⊢ (𝜑 → (𝑅‘𝑌) ∈ (((1^st ‘𝐹)‘𝑌)(Hom ‘𝐶)𝑋))
32	15, 28, 24, 13, 30, 6, 16, 31	catlid 17771	. . 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 1570 ∈ wcel 2145 ⟨cop 4590 ‘cfv 6533 (class class class)co 7413 1^st c1st 7984 2^nd c2nd 7985 Basecbs 17301 Hom chom 17353 compcco 17354 Idccid 17753 Nat cnat 18033 Δ_funccdiag 18300

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-struct 17239 df-slot 17274 df-ndx 17286 df-base 17302 df-hom 17366 df-cco 17367 df-cat 17756 df-cid 17757 df-func 17947 df-nat 18035 df-xpc 18260 df-1stf 18261 df-curf 18302 df-diag 18304

This theorem is used by: (None)

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