Theorem concom 50361

Description: A cone to a diagram commutes with the diagram. (Contributed by Zhi Wang, 13-Nov-2025.)

Hypotheses

Ref	Expression
islmd.l	⊢ 𝐿 = (𝐶Δfunc𝐷)
islmd.a	⊢ 𝐴 = (Base‘𝐶)
islmd.n	⊢ 𝑁 = (𝐷 Nat 𝐶)
islmd.b	⊢ 𝐵 = (Base‘𝐷)
concl.k	⊢ 𝐾 = ((1st ‘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(2nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘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 18011	. . 3 ⊢ (𝜑 → 𝑅 ∈ (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩𝑁⟨(1st ‘𝐹), (2nd ‘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 18015	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍))((𝑌(2nd ‘𝐾)𝑍)‘𝑀)) = (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)))
11		islmd.l	. . . . . . 7 ⊢ 𝐿 = (𝐶Δfunc𝐷)
12	1, 3	natrcl3 49923	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
13	12	funcrcl3 49778	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49777	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ Cat)
15		islmd.a	. . . . . . 7 ⊢ 𝐴 = (Base‘𝐶)
16		concl.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
17		concl.k	. . . . . . 7 ⊢ 𝐾 = ((1st ‘𝐿)‘𝑋)
18	11, 13, 14, 15, 16, 17, 4, 7	diag11 18299	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18299	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4848	. . . . 5 ⊢ (𝜑 → ⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7426	. . . 4 ⊢ (𝜑 → (⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1st ‘𝐹)‘𝑍)))
22		eqidd 2770	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) = (𝑅‘𝑍))
23		eqid 2769	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
24	11, 13, 14, 15, 16, 17, 4, 7, 5, 23, 8, 9	diag12 18300	. . . 4 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7432	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍))((𝑌(2nd ‘𝐾)𝑍)‘𝑀)) = ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)))
26		eqid 2769	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17923	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7081	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 50359	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17740	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2804	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍))((𝑌(2nd ‘𝐾)𝑍)‘𝑀)) = (𝑅‘𝑍))
32	18	opeq1d 4846	. . . 4 ⊢ (𝜑 → ⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7426	. . 3 ⊢ (𝜑 → (⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍)) = (⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍)))
34	33	oveqd 7428	. 2 ⊢ (𝜑 → (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)) = (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)))
35	10, 31, 34	3eqtr3d 2812	1 ⊢ (𝜑 → (𝑅‘𝑍) = (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ⟨cop 4598  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17269  Hom chom 17321  compcco 17322  Idccid 17721   Nat cnat 18001  Δ_funccdiag 18268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-fz 13536  df-struct 17207  df-slot 17242  df-ndx 17254  df-base 17270  df-hom 17334  df-cco 17335  df-cat 17724  df-cid 17725  df-func 17915  df-nat 18003  df-xpc 18228  df-1stf 18229  df-curf 18270  df-diag 18272

This theorem is referenced by: (None)