Theorem concom 49662

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 17848	. . 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 17852	. 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 49224	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
13	12	funcrcl3 49079	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49078	. . . . . . 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 18136	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18136	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4830	. . . . 5 ⊢ (𝜑 → ⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7355	. . . 4 ⊢ (𝜑 → (⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1st ‘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 18137	. . . 4 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7361	. . 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 2729	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17760	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7012	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 49660	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17577	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2764	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍))((𝑌(2nd ‘𝐾)𝑍)‘𝑀)) = (𝑅‘𝑍))
32	18	opeq1d 4828	. . . 4 ⊢ (𝜑 → ⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7355	. . 3 ⊢ (𝜑 → (⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍)) = (⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍)))
34	33	oveqd 7357	. 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 2772	1 ⊢ (𝜑 → (𝑅‘𝑍) = (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4579  ‘cfv 6476  (class class class)co 7340  1^st c1st 7913  2^nd c2nd 7914  Basecbs 17107  Hom chom 17159  compcco 17160  Idccid 17558   Nat cnat 17838  Δ_funccdiag 18105

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 5214  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5367  ax-un 7662  ax-cnex 11053  ax-resscn 11054  ax-1cn 11055  ax-icn 11056  ax-addcl 11057  ax-addrcl 11058  ax-mulcl 11059  ax-mulrcl 11060  ax-mulcom 11061  ax-addass 11062  ax-mulass 11063  ax-distr 11064  ax-i2m1 11065  ax-1ne0 11066  ax-1rid 11067  ax-rnegex 11068  ax-rrecex 11069  ax-cnre 11070  ax-pre-lttri 11071  ax-pre-lttrn 11072  ax-pre-ltadd 11073  ax-pre-mulgt0 11074

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 3343  df-reu 3344  df-rab 3393  df-v 3435  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  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 4940  df-br 5089  df-opab 5151  df-mpt 5170  df-tr 5196  df-id 5508  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5566  df-we 5568  df-xp 5619  df-rel 5620  df-cnv 5621  df-co 5622  df-dm 5623  df-rn 5624  df-res 5625  df-ima 5626  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7297  df-ov 7343  df-oprab 7344  df-mpo 7345  df-om 7791  df-1st 7915  df-2nd 7916  df-frecs 8205  df-wrecs 8236  df-recs 8285  df-rdg 8323  df-1o 8379  df-er 8616  df-map 8746  df-ixp 8816  df-en 8864  df-dom 8865  df-sdom 8866  df-fin 8867  df-pnf 11139  df-mnf 11140  df-xr 11141  df-ltxr 11142  df-le 11143  df-sub 11337  df-neg 11338  df-nn 12117  df-2 12179  df-3 12180  df-4 12181  df-5 12182  df-6 12183  df-7 12184  df-8 12185  df-9 12186  df-n0 12373  df-z 12460  df-dec 12580  df-uz 12724  df-fz 13399  df-struct 17045  df-slot 17080  df-ndx 17092  df-base 17108  df-hom 17172  df-cco 17173  df-cat 17561  df-cid 17562  df-func 17752  df-nat 17840  df-xpc 18065  df-1stf 18066  df-curf 18107  df-diag 18109

This theorem is referenced by: (None)