Theorem concom 50469

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 18017	. . 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 18021	. 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 50031	. . . . . . . 8 ⊢ (𝜑 → (1st ‘𝐹)(𝐷 Func 𝐶)(2nd ‘𝐹))
13	12	funcrcl3 49886	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49885	. . . . . . 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 18305	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18305	. . . . . 6 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4845	. . . . 5 ⊢ (𝜑 → ⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7427	. . . 4 ⊢ (𝜑 → (⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1st ‘𝐹)‘𝑍)))
22		eqidd 2763	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) = (𝑅‘𝑍))
23		eqid 2762	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
24	11, 13, 14, 15, 16, 17, 4, 7, 5, 23, 8, 9	diag12 18306	. . . 4 ⊢ (𝜑 → ((𝑌(2nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7433	. . 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 2762	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17929	. . . . 5 ⊢ (𝜑 → (1st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7080	. . . 4 ⊢ (𝜑 → ((1st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 50467	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17746	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2797	. 2 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐾)‘𝑍)⟩ · ((1st ‘𝐹)‘𝑍))((𝑌(2nd ‘𝐾)𝑍)‘𝑀)) = (𝑅‘𝑍))
32	18	opeq1d 4843	. . . 4 ⊢ (𝜑 → ⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7427	. . 3 ⊢ (𝜑 → (⟨((1st ‘𝐾)‘𝑌), ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍)) = (⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍)))
34	33	oveqd 7429	. 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 2805	1 ⊢ (𝜑 → (𝑅‘𝑍) = (((𝑌(2nd ‘𝐹)𝑍)‘𝑀)(⟨𝑋, ((1st ‘𝐹)‘𝑌)⟩ · ((1st ‘𝐹)‘𝑍))(𝑅‘𝑌)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ⟨cop 4594  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983  Basecbs 17275  Hom chom 17327  compcco 17328  Idccid 17727   Nat cnat 18007  Δ_funccdiag 18274

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-7 12314  df-8 12315  df-9 12316  df-n0 12511  df-z 12598  df-dec 12718  df-uz 12869  df-fz 13542  df-struct 17213  df-slot 17248  df-ndx 17260  df-base 17276  df-hom 17340  df-cco 17341  df-cat 17730  df-cid 17731  df-func 17921  df-nat 18009  df-xpc 18234  df-1stf 18235  df-curf 18276  df-diag 18278

This theorem is used by: (None)