concom - Mathbox for Zhi Wang

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Theorem concom 49763

Description: A 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)
concom.r	⊢ (𝜑 → 𝑅 ∈ (𝐾𝑁𝐹))

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

**Proof of Theorem concom**
Step	Hyp	Ref	Expression
1		islmd.n	. . 3 ⊢ 𝑁 = (𝐷 Nat 𝐶)
2		concom.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ (𝐾𝑁𝐹))
3	1, 2	nat1st2nd 17861	. . 3 ⊢ (𝜑 → 𝑅 ∈ (⟨(1^st ‘𝐾), (2^nd ‘𝐾)⟩𝑁⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩))
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 17865	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍))((𝑌(2^nd ‘𝐾)𝑍)‘𝑀)) = (((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)(⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ · ((1^st ‘𝐹)‘𝑍))(𝑅‘𝑌)))
11		islmd.l	. . . . . . 7 ⊢ 𝐿 = (𝐶Δ_func𝐷)
12	1, 3	natrcl3 49325	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 49180	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49179	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
15		islmd.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝐶)
16		concl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
17		concl.k	. . . . . . 7 ⊢ 𝐾 = ((1^st ‘𝐿)‘𝑋)
18	11, 13, 14, 15, 16, 17, 4, 7	diag11 18149	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18149	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4830	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7361	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍)))
22		eqidd 2732	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) = (𝑅‘𝑍))
23		eqid 2731	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
24	11, 13, 14, 15, 16, 17, 4, 7, 5, 23, 8, 9	diag12 18150	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7367	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍))((𝑌(2^nd ‘𝐾)𝑍)‘𝑀)) = ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)))
26		eqid 2731	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17773	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7018	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 49761	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1^st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17590	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2766	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍))((𝑌(2^nd ‘𝐾)𝑍)‘𝑀)) = (𝑅‘𝑍))
32	18	opeq1d 4828	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7361	. . 3 ⊢ (𝜑 → (⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ · ((1^st ‘𝐹)‘𝑍)) = (⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩ · ((1^st ‘𝐹)‘𝑍)))
34	33	oveqd 7363	. 2 ⊢ (𝜑 → (((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)(⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ · ((1^st ‘𝐹)‘𝑍))(𝑅‘𝑌)) = (((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩ · ((1^st ‘𝐹)‘𝑍))(𝑅‘𝑌)))
35	10, 31, 34	3eqtr3d 2774	1 ⊢ (𝜑 → (𝑅‘𝑍) = (((𝑌(2^nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩ · ((1^st ‘𝐹)‘𝑍))(𝑅‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 ⟨cop 4579 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 Basecbs 17120 Hom chom 17172 compcco 17173 Idccid 17571 Nat cnat 17851 Δ_funccdiag 18118

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-hom 17185 df-cco 17186 df-cat 17574 df-cid 17575 df-func 17765 df-nat 17853 df-xpc 18078 df-1stf 18079 df-curf 18120 df-diag 18122

This theorem is referenced by: (None)

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