Theorem oppccofval 16199

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 6131	. . . . . 6 ⊢ (𝑋 ∈ (Base‘𝐶) → 𝐶 ∈ V)
3		oppcco.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eleq2s 2706	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝐶 ∈ V)
5		eqid 2610	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		oppcco.c	. . . . . 6 ⊢ · = (comp‘𝐶)
7		oppcco.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
8	3, 5, 6, 7	oppcval 16196	. . . . 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 18	. . . 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 6107	. . 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 6577	. . . 4 ⊢ (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
12		fvex 6113	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
13	3, 12	eqeltri 2684	. . . . . 6 ⊢ 𝐵 ∈ V
14	13, 13	xpex 6860	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
15	14, 13	mpt2ex 7136	. . . 4 ⊢ (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))) ∈ V
16		ccoid 15900	. . . . 5 ⊢ comp = Slot (comp‘ndx)
17	16	setsid 15742	. . . 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)))⟩)))
18	11, 15, 17	mp2an 704	. . 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)))⟩))
19	10, 18	syl6eqr 2662	. 2 ⊢ (𝜑 → (comp‘𝑂) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))))
20		simprr 792	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
21		simprl 790	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑢 = ⟨𝑋, 𝑌⟩)
22	21	fveq2d 6107	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑢) = (2nd ‘⟨𝑋, 𝑌⟩))
23	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑋 ∈ 𝐵)
24		oppcco.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑌 ∈ 𝐵)
26		op2ndg 7072	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
27	23, 25, 26	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
28	22, 27	eqtrd 2644	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘𝑢) = 𝑌)
29	20, 28	opeq12d 4348	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2nd ‘𝑢)⟩ = ⟨𝑍, 𝑌⟩)
30	21	fveq2d 6107	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑢) = (1st ‘⟨𝑋, 𝑌⟩))
31		op1stg 7071	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
32	23, 25, 31	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
33	30, 32	eqtrd 2644	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘𝑢) = 𝑋)
34	29, 33	oveq12d 6567	. . 3 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)) = (⟨𝑍, 𝑌⟩ · 𝑋))
35	34	tposeqd 7242	. 2 ⊢ ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))
36		opelxpi 5072	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
37	1, 24, 36	syl2anc 691	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
38		oppcco.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
39		ovex 6577	. . . 4 ⊢ (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
40	39	tposex 7273	. . 3 ⊢ tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V
41	40	a1i 11	. 2 ⊢ (𝜑 → tpos (⟨𝑍, 𝑌⟩ · 𝑋) ∈ V)
42	19, 35, 37, 38, 41	ovmpt2d 6686	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌⟩ · 𝑋))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058  tpos ctpos 7238  ndxcnx 15692   sSet csts 15693  Basecbs 15695  Hom chom 15779  compcco 15780  oppCatcoppc 16194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-dec 11370  df-ndx 15698  df-slot 15699  df-sets 15701  df-cco 15794  df-oppc 16195

This theorem is referenced by:  oppcco  16200