concom - Mathbox for Zhi Wang

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Theorem concom 49975

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 17882	. . 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 17886	. 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 49537	. . . . . . . 8 ⊢ (𝜑 → (1^st ‘𝐹)(𝐷 Func 𝐶)(2^nd ‘𝐹))
13	12	funcrcl3 49392	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	12	funcrcl2 49391	. . . . . . 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 18170	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑌) = 𝑋)
19	11, 13, 14, 15, 16, 17, 4, 8	diag11 18170	. . . . . 6 ⊢ (𝜑 → ((1^st ‘𝐾)‘𝑍) = 𝑋)
20	18, 19	opeq12d 4838	. . . . 5 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ = ⟨𝑋, 𝑋⟩)
21	20	oveq1d 7375	. . . 4 ⊢ (𝜑 → (⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐾)‘𝑍)⟩ · ((1^st ‘𝐹)‘𝑍)) = (⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍)))
22		eqidd 2738	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) = (𝑅‘𝑍))
23		eqid 2737	. . . . 5 ⊢ (Id‘𝐶) = (Id‘𝐶)
24	11, 13, 14, 15, 16, 17, 4, 7, 5, 23, 8, 9	diag12 18171	. . . 4 ⊢ (𝜑 → ((𝑌(2^nd ‘𝐾)𝑍)‘𝑀) = ((Id‘𝐶)‘𝑋))
25	21, 22, 24	oveq123d 7381	. . 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 2737	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
27	4, 15, 12	funcf1 17794	. . . . 5 ⊢ (𝜑 → (1^st ‘𝐹):𝐵⟶𝐴)
28	27, 8	ffvelcdmd 7032	. . . 4 ⊢ (𝜑 → ((1^st ‘𝐹)‘𝑍) ∈ 𝐴)
29	11, 15, 1, 4, 17, 16, 8, 26, 2	concl 49973	. . . 4 ⊢ (𝜑 → (𝑅‘𝑍) ∈ (𝑋(Hom ‘𝐶)((1^st ‘𝐹)‘𝑍)))
30	15, 26, 23, 13, 16, 6, 28, 29	catrid 17611	. . 3 ⊢ (𝜑 → ((𝑅‘𝑍)(⟨𝑋, 𝑋⟩ · ((1^st ‘𝐹)‘𝑍))((Id‘𝐶)‘𝑋)) = (𝑅‘𝑍))
31	25, 30	eqtrd 2772	. 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 4836	. . . 4 ⊢ (𝜑 → ⟨((1^st ‘𝐾)‘𝑌), ((1^st ‘𝐹)‘𝑌)⟩ = ⟨𝑋, ((1^st ‘𝐹)‘𝑌)⟩)
33	32	oveq1d 7375	. . 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 7377	. 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 2780	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 1542 ∈ wcel 2114 ⟨cop 4587 ‘cfv 6493 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17140 Hom chom 17192 compcco 17193 Idccid 17592 Nat cnat 17872 Δ_funccdiag 18139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-fz 13428 df-struct 17078 df-slot 17113 df-ndx 17125 df-base 17141 df-hom 17205 df-cco 17206 df-cat 17595 df-cid 17596 df-func 17786 df-nat 17874 df-xpc 18099 df-1stf 18100 df-curf 18141 df-diag 18143

This theorem is referenced by: (None)

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