Theorem oppccofval 17896

Description: Composition in the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
oppcco.b	⊢ 𝐵 = (Base‘𝐶)
oppcco.c	⊢ · = (comp‘𝐶)
oppcco.o	⊢ 𝑂 = (oppCat‘𝐶)
oppcco.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
oppcco.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
oppcco.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
oppccofval	⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))

Proof of Theorem oppccofval

Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppcco.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		elfvex 6921	. . . . . 6 ⊢ (𝑋 ∈ (Base‘𝐶) → 𝐶 ∈ V)
3		oppcco.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eleq2s 2878	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝐶 ∈ V)
5		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		oppcco.c	. . . . . 6 ⊢ · = (comp‘𝐶)
7		oppcco.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
8	3, 5, 6, 7	oppcval 17893	. . . . 5 ⊢ (𝐶 ∈ V → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))
9	1, 4, 8	3syl 19	. . . 4 ⊢ (𝜑 → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))
10	9	fveq2d 6890	. . 3 ⊢ (𝜑 → (comp‘𝑂) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩)))
11		ovex 7454	. . . 4 ⊢ (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
12	3	fvexi 6900	. . . . . 6 ⊢ 𝐵 ∈ V
13	12, 12	xpex 7768	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
14	13, 12	mpoex 8093	. . . 4 ⊢ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))) ∈ V
15		ccoid 17591	. . . . 5 ⊢ comp = Slot (comp‘ndx)
16	15	setsid 17391	. . . 4 ⊢ (((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V ∧ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))) ∈ V) → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩)))
17	11, 14, 16	mp2an 705	. . 3 ⊢ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))
18	10, 17	eqtr4di 2813	. 2 ⊢ (𝜑 → (comp‘𝑂) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))))
19		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
20		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑢 = ⟨𝑋, 𝑌⟩)
21	20	fveq2d 6890	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑢) = (2nd ‘⟨𝑋, 𝑌⟩))
22		oppcco.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
23	22	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑌 ∈ 𝐵)
24		op2ndg 8015	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
25	1, 23, 24	syl2an2r 698	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
26	21, 25	eqtrd 2795	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑢) = 𝑌)
27	19, 26	opeq12d 4847	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2nd ‘𝑢)⟩ = ⟨𝑍, 𝑌⟩)
28	20	fveq2d 6890	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑢) = (1st ‘⟨𝑋, 𝑌⟩))
29		op1stg 8014	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
30	1, 23, 29	syl2an2r 698	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
31	28, 30	eqtrd 2795	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑢) = 𝑋)
32	27, 31	oveq12d 7439	. . 3 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)) = (⟨𝑍, 𝑌⟩ · 𝑋))
33	32	tposeqd 8250	. 2 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))
34	1, 22	opelxpd 5689	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
35		oppcco.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
36		ovex 7454	. . . 4 ⊢ (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
37	36	tposex 8281	. . 3 ⊢ tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
38	37	a1i 11	. 2 ⊢ (𝜑 → tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V)
39	18, 33, 34, 35, 38	ovmpod 7573	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   × cxp 5648  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  1^st c1st 8000  2^nd c2nd 8001  tpos ctpos 8246   sSet csts 17347  ndxcnx 17377  Basecbs 17393  Hom chom 17445  compcco 17446  oppCatcoppc 17891

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-tpos 8247  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-pnf 11349  df-mnf 11350  df-ltxr 11352  df-nn 12340  df-2 12409  df-3 12410  df-4 12411  df-5 12412  df-6 12413  df-7 12414  df-8 12415  df-9 12416  df-n0 12611  df-dec 12819  df-sets 17348  df-slot 17366  df-ndx 17378  df-cco 17459  df-oppc 17892

This theorem is used by:  oppcco  17897  oppgoppcco  50823