concom - Mathbox for Zhi Wang

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Theorem concom 49656

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 17923	. . 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 17927	. 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 49218	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 49073	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49072	. . . . . . 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 18211	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18211	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4848	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7405	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍)))
22		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) = (𝑅‘𝑍))
23		eqid 2730	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
24	11, 13, 14, 15, 16, 17, 4, 7, 5, 23, 8, 9	diag12 18212	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7411	. . 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 2730	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17835	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7060	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 49654	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1^st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17652	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2765	. 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 4846	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7405	. . 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 7407	. 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 2773	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 4598 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17186 Hom chom 17238 compcco 17239 Idccid 17633 Nat cnat 17913 Δ_funccdiag 18180

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-hom 17251 df-cco 17252 df-cat 17636 df-cid 17637 df-func 17827 df-nat 17915 df-xpc 18140 df-1stf 18141 df-curf 18182 df-diag 18184

This theorem is referenced by: (None)

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