Theorem oppccofval 17804

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 6913	. . . . . 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 17801	. . . . 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 6882	. . 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 7446	. . . 4 ⊢ (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
12	3	fvexi 6892	. . . . . 6 ⊢ 𝐵 ∈ V
13	12, 12	xpex 7752	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
14	13, 12	mpoex 8078	. . . 4 ⊢ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))) ∈ V
15		ccoid 17499	. . . . 5 ⊢ comp = Slot (comp‘ndx)
16	15	setsid 17299	. . . 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 6882	. . . . . 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 7999	. . . . . . 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 4841	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2nd ‘𝑢)⟩ = ⟨𝑍, 𝑌⟩)
28	20	fveq2d 6882	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑢) = (1st ‘⟨𝑋, 𝑌⟩))
29		op1stg 7998	. . . . . 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 7431	. . 3 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)) = (⟨𝑍, 𝑌⟩ · 𝑋))
33	32	tposeqd 8227	. 2 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))
34	1, 22	opelxpd 5694	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
35		oppcco.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
36		ovex 7446	. . . 4 ⊢ (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
37	36	tposex 8258	. . 3 ⊢ tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
38	37	a1i 11	. 2 ⊢ (𝜑 → tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V)
39	18, 33, 34, 35, 38	ovmpod 7565	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ⟨cop 4590   × cxp 5653  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  1^st c1st 7984  2^nd c2nd 7985  tpos ctpos 8223   sSet csts 17255  ndxcnx 17285  Basecbs 17301  Hom chom 17353  compcco 17354  oppCatcoppc 17799

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11269  df-mnf 11270  df-ltxr 11272  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-dec 12737  df-sets 17256  df-slot 17274  df-ndx 17286  df-cco 17367  df-oppc 17800

This theorem is used by:  oppcco  17805  oppgoppcco  50517