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Theorem oppccofval 16977
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 6685 . . . . . 6 (𝑋 ∈ (Base‘𝐶) → 𝐶 ∈ V)
3 oppcco.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3eleq2s 2932 . . . . 5 (𝑋𝐵𝐶 ∈ V)
5 eqid 2822 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
6 oppcco.c . . . . . 6 · = (comp‘𝐶)
7 oppcco.o . . . . . 6 𝑂 = (oppCat‘𝐶)
83, 5, 6, 7oppcval 16974 . . . . 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 6656 . . 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 7173 . . . 4 (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
123fvexi 6666 . . . . . 6 𝐵 ∈ V
1312, 12xpex 7461 . . . . 5 (𝐵 × 𝐵) ∈ V
1413, 12mpoex 7764 . . . 4 (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))) ∈ V
15 ccoid 16681 . . . . 5 comp = Slot (comp‘ndx)
1615setsid 16529 . . . 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 691 . . 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, 17eqtr4di 2875 . 2 (𝜑 → (comp‘𝑂) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))))
19 simprr 772 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
20 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑢 = ⟨𝑋, 𝑌⟩)
2120fveq2d 6656 . . . . . 6 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑢) = (2nd ‘⟨𝑋, 𝑌⟩))
22 oppcco.y . . . . . . . 8 (𝜑𝑌𝐵)
2322adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑌𝐵)
24 op2ndg 7688 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
251, 23, 24syl2an2r 684 . . . . . 6 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2621, 25eqtrd 2857 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑢) = 𝑌)
2719, 26opeq12d 4786 . . . 4 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2nd𝑢)⟩ = ⟨𝑍, 𝑌⟩)
2820fveq2d 6656 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑢) = (1st ‘⟨𝑋, 𝑌⟩))
29 op1stg 7687 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
301, 23, 29syl2an2r 684 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3128, 30eqtrd 2857 . . . 4 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑢) = 𝑋)
3227, 31oveq12d 7158 . . 3 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)) = (⟨𝑍, 𝑌· 𝑋))
3332tposeqd 7882 . 2 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)) = tpos (⟨𝑍, 𝑌· 𝑋))
341, 22opelxpd 5570 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
35 oppcco.z . 2 (𝜑𝑍𝐵)
36 ovex 7173 . . . 4 (⟨𝑍, 𝑌· 𝑋) ∈ V
3736tposex 7913 . . 3 tpos (⟨𝑍, 𝑌· 𝑋) ∈ V
3837a1i 11 . 2 (𝜑 → tpos (⟨𝑍, 𝑌· 𝑋) ∈ V)
3918, 33, 34, 35, 38ovmpod 7286 1 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌· 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  cop 4545   × cxp 5530  cfv 6334  (class class class)co 7140  cmpo 7142  1st c1st 7673  2nd c2nd 7674  tpos ctpos 7878  ndxcnx 16471   sSet csts 16472  Basecbs 16474  Hom chom 16567  compcco 16568  oppCatcoppc 16972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-ltxr 10669  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-dec 12087  df-ndx 16477  df-slot 16478  df-sets 16481  df-cco 16581  df-oppc 16973
This theorem is referenced by:  oppcco  16978
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