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Theorem oppccofval 16765
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 (𝜑𝑋𝐵)
oppcco.y (𝜑𝑌𝐵)
oppcco.z (𝜑𝑍𝐵)
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.
StepHypRef Expression
1 oppcco.x . . . . 5 (𝜑𝑋𝐵)
2 elfvex 6482 . . . . . 6 (𝑋 ∈ (Base‘𝐶) → 𝐶 ∈ V)
3 oppcco.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3eleq2s 2877 . . . . 5 (𝑋𝐵𝐶 ∈ V)
5 eqid 2778 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
6 oppcco.c . . . . . 6 · = (comp‘𝐶)
7 oppcco.o . . . . . 6 𝑂 = (oppCat‘𝐶)
83, 5, 6, 7oppcval 16762 . . . . 5 (𝐶 ∈ V → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
91, 4, 83syl 18 . . . 4 (𝜑𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
109fveq2d 6452 . . 3 (𝜑 → (comp‘𝑂) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩)))
11 ovex 6956 . . . 4 (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
123fvexi 6462 . . . . . 6 𝐵 ∈ V
1312, 12xpex 7242 . . . . 5 (𝐵 × 𝐵) ∈ V
1413, 12mpt2ex 7529 . . . 4 (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))) ∈ V
15 ccoid 16467 . . . . 5 comp = Slot (comp‘ndx)
1615setsid 16314 . . . 4 (((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V ∧ (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))) ∈ V) → (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩)))
1711, 14, 16mp2an 682 . . 3 (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))) = (comp‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
1810, 17syl6eqr 2832 . 2 (𝜑 → (comp‘𝑂) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))))
19 simprr 763 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
20 simprl 761 . . . . . . 7 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑢 = ⟨𝑋, 𝑌⟩)
2120fveq2d 6452 . . . . . 6 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑢) = (2nd ‘⟨𝑋, 𝑌⟩))
22 oppcco.y . . . . . . . 8 (𝜑𝑌𝐵)
2322adantr 474 . . . . . . 7 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑌𝐵)
24 op2ndg 7460 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
251, 23, 24syl2an2r 675 . . . . . 6 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2621, 25eqtrd 2814 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑢) = 𝑌)
2719, 26opeq12d 4646 . . . 4 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2nd𝑢)⟩ = ⟨𝑍, 𝑌⟩)
2820fveq2d 6452 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑢) = (1st ‘⟨𝑋, 𝑌⟩))
29 op1stg 7459 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
301, 23, 29syl2an2r 675 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3128, 30eqtrd 2814 . . . 4 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑢) = 𝑋)
3227, 31oveq12d 6942 . . 3 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)) = (⟨𝑍, 𝑌· 𝑋))
3332tposeqd 7639 . 2 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)) = tpos (⟨𝑍, 𝑌· 𝑋))
341, 22opelxpd 5395 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
35 oppcco.z . 2 (𝜑𝑍𝐵)
36 ovex 6956 . . . 4 (⟨𝑍, 𝑌· 𝑋) ∈ V
3736tposex 7670 . . 3 tpos (⟨𝑍, 𝑌· 𝑋) ∈ V
3837a1i 11 . 2 (𝜑 → tpos (⟨𝑍, 𝑌· 𝑋) ∈ V)
3918, 33, 34, 35, 38ovmpt2d 7067 1 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌· 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  cop 4404   × cxp 5355  cfv 6137  (class class class)co 6924  cmpt2 6926  1st c1st 7445  2nd c2nd 7446  tpos ctpos 7635  ndxcnx 16256   sSet csts 16257  Basecbs 16259  Hom chom 16353  compcco 16354  oppCatcoppc 16760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-cnex 10330  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-tpos 7636  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-ltxr 10418  df-nn 11379  df-2 11442  df-3 11443  df-4 11444  df-5 11445  df-6 11446  df-7 11447  df-8 11448  df-9 11449  df-n0 11647  df-dec 11850  df-ndx 16262  df-slot 16263  df-sets 16266  df-cco 16367  df-oppc 16761
This theorem is referenced by:  oppcco  16766
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