concom - Mathbox for Zhi Wang

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Theorem concom 49645

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 17896	. . 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 17900	. 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 49207	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 49062	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49061	. . . . . . 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 18184	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18184	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4841	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7384	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍)))
22		eqidd 2730	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) = (𝑅‘𝑍))
23		eqid 2729	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
24	11, 13, 14, 15, 16, 17, 4, 7, 5, 23, 8, 9	diag12 18185	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7390	. . 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 2729	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17808	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7039	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 49643	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1^st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17625	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2764	. 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 4839	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7384	. . 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 7386	. 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 2772	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 1540 ∈ wcel 2109 ⟨cop 4591 ‘cfv 6499 (class class class)co 7369 1^st c1st 7945 2^nd c2nd 7946 Basecbs 17155 Hom chom 17207 compcco 17208 Idccid 17606 Nat cnat 17886 Δ_funccdiag 18153

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-hom 17220 df-cco 17221 df-cat 17609 df-cid 17610 df-func 17800 df-nat 17888 df-xpc 18113 df-1stf 18114 df-curf 18155 df-diag 18157

This theorem is referenced by: (None)

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