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Theorem oppccofval 17668

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 6922	. . . . . 6 ⊢ (𝑋 ∈ (Base‘𝐶) → 𝐶 ∈ V)
3		oppcco.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eleq2s 2845	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝐶 ∈ V)
5		eqid 2726	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		oppcco.c	. . . . . 6 ⊢ · = (comp‘𝐶)
7		oppcco.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
8	3, 5, 6, 7	oppcval 17664	. . . . 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 6888	. . 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 7437	. . . 4 ⊢ (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
12	3	fvexi 6898	. . . . . 6 ⊢ 𝐵 ∈ V
13	12, 12	xpex 7736	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
14	13, 12	mpoex 8062	. . . 4 ⊢ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))) ∈ V
15		ccoid 17366	. . . . 5 ⊢ comp = Slot (comp‘ndx)
16	15	setsid 17148	. . . 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 689	. . 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 2784	. 2 ⊢ (𝜑 → (comp‘𝑂) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))))
19		simprr 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
20		simprl 768	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑢 = ⟨𝑋, 𝑌⟩)
21	20	fveq2d 6888	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘𝑢) = (2^nd ‘⟨𝑋, 𝑌⟩))
22		oppcco.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑌 ∈ 𝐵)
24		op2ndg 7984	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
25	1, 23, 24	syl2an2r 682	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
26	21, 25	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2^nd ‘𝑢) = 𝑌)
27	19, 26	opeq12d 4876	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2^nd ‘𝑢)⟩ = ⟨𝑍, 𝑌⟩)
28	20	fveq2d 6888	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘𝑢) = (1^st ‘⟨𝑋, 𝑌⟩))
29		op1stg 7983	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
30	1, 23, 29	syl2an2r 682	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
31	28, 30	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1^st ‘𝑢) = 𝑋)
32	27, 31	oveq12d 7422	. . 3 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)) = (⟨𝑍, 𝑌⟩ · 𝑋))
33	32	tposeqd 8212	. 2 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))
34	1, 22	opelxpd 5708	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
35		oppcco.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
36		ovex 7437	. . . 4 ⊢ (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
37	36	tposex 8243	. . 3 ⊢ tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
38	37	a1i 11	. 2 ⊢ (𝜑 → tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V)
39	18, 33, 34, 35, 38	ovmpod 7555	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 × cxp 5667 ‘cfv 6536 (class class class)co 7404 ∈ cmpo 7406 1^st c1st 7969 2^nd c2nd 7970 tpos ctpos 8208 sSet csts 17103 ndxcnx 17133 Basecbs 17151 Hom chom 17215 compcco 17216 oppCatcoppc 17662

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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-ltxr 11254 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-dec 12679 df-sets 17104 df-slot 17122 df-ndx 17134 df-cco 17229 df-oppc 17663

This theorem is referenced by: oppcco 17669

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