concom - Mathbox for Zhi Wang

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Theorem concom 50167

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 17916	. . 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 17920	. 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 49729	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 49584	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49583	. . . . . . 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 18204	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18204	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4815	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7375	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍)))
22		eqidd 2742	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) = (𝑅‘𝑍))
23		eqid 2741	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
24	11, 13, 14, 15, 16, 17, 4, 7, 5, 23, 8, 9	diag12 18205	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7381	. . 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 2741	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17828	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 50165	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1^st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17645	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2776	. 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 4813	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7375	. . 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 7377	. 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 2784	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 1548 ∈ wcel 2121 ⟨cop 4564 ‘cfv 6489 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17174 Hom chom 17226 compcco 17227 Idccid 17626 Nat cnat 17906 Δ_funccdiag 18173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-fz 13457 df-struct 17112 df-slot 17147 df-ndx 17159 df-base 17175 df-hom 17239 df-cco 17240 df-cat 17629 df-cid 17630 df-func 17820 df-nat 17908 df-xpc 18133 df-1stf 18134 df-curf 18175 df-diag 18177

This theorem is referenced by: (None)

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