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Theorem oppccofval 17702

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 6938	. . . . . 6 ⊢ (𝑋 ∈ (Base‘𝐶) → 𝐶 ∈ V)
3		oppcco.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eleq2s 2846	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝐶 ∈ V)
5		eqid 2727	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		oppcco.c	. . . . . 6 ⊢ · = (comp‘𝐶)
7		oppcco.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
8	3, 5, 6, 7	oppcval 17698	. . . . 5 ⊢ (𝐶 ∈ V → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩))
9	1, 4, 8	3syl 18	. . . 4 ⊢ (𝜑 → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩))
10	9	fveq2d 6904	. . 3 ⊢ (𝜑 → (comp‘𝑂) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩)))
11		ovex 7457	. . . 4 ⊢ (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
12	3	fvexi 6914	. . . . . 6 ⊢ 𝐵 ∈ V
13	12, 12	xpex 7759	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
14	13, 12	mpoex 8088	. . . 4 ⊢ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))) ∈ V
15		ccoid 17400	. . . . 5 ⊢ comp = Slot (comp‘ndx)
16	15	setsid 17182	. . . 4 ⊢ (((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V ∧ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))) ∈ V) → (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩)))
17	11, 14, 16	mp2an 690	. . 3 ⊢ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩))
18	10, 17	eqtr4di 2785	. 2 ⊢ (𝜑 → (comp‘𝑂) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))))
19		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
20		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑢 = ⟨𝑋, 𝑌⟩)
21	20	fveq2d 6904	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘𝑢) = (2^nd ‘⟨𝑋, 𝑌⟩))
22		oppcco.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
23	22	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑌 ∈ 𝐵)
24		op2ndg 8010	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
25	1, 23, 24	syl2an2r 683	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
26	21, 25	eqtrd 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘𝑢) = 𝑌)
27	19, 26	opeq12d 4884	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2^nd ‘𝑢)⟩ = ⟨𝑍, 𝑌⟩)
28	20	fveq2d 6904	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘𝑢) = (1^st ‘⟨𝑋, 𝑌⟩))
29		op1stg 8009	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
30	1, 23, 29	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
31	28, 30	eqtrd 2767	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘𝑢) = 𝑋)
32	27, 31	oveq12d 7442	. . 3 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)) = (⟨𝑍, 𝑌⟩ · 𝑋))
33	32	tposeqd 8239	. 2 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))
34	1, 22	opelxpd 5719	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
35		oppcco.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
36		ovex 7457	. . . 4 ⊢ (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
37	36	tposex 8270	. . 3 ⊢ tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
38	37	a1i 11	. 2 ⊢ (𝜑 → tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V)
39	18, 33, 34, 35, 38	ovmpod 7577	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ⟨cop 4636 × cxp 5678 ‘cfv 6551 (class class class)co 7424 ∈ cmpo 7426 1^st c1st 7995 2^nd c2nd 7996 tpos ctpos 8235 sSet csts 17137 ndxcnx 17167 Basecbs 17185 Hom chom 17249 compcco 17250 oppCatcoppc 17696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-tpos 8236 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-ltxr 11289 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-dec 12714 df-sets 17138 df-slot 17156 df-ndx 17168 df-cco 17263 df-oppc 17697

This theorem is referenced by: oppcco 17703

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