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Theorem oppccofval 16575
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 6380 . . . . . 6 (𝑋 ∈ (Base‘𝐶) → 𝐶 ∈ V)
3 oppcco.b . . . . . 6 𝐵 = (Base‘𝐶)
42, 3eleq2s 2855 . . . . 5 (𝑋𝐵𝐶 ∈ V)
5 eqid 2758 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
6 oppcco.c . . . . . 6 · = (comp‘𝐶)
7 oppcco.o . . . . . 6 𝑂 = (oppCat‘𝐶)
83, 5, 6, 7oppcval 16572 . . . . 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 6354 . . 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 6839 . . . 4 (𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) ∈ V
12 fvex 6360 . . . . . . 7 (Base‘𝐶) ∈ V
133, 12eqeltri 2833 . . . . . 6 𝐵 ∈ V
1413, 13xpex 7125 . . . . 5 (𝐵 × 𝐵) ∈ V
1514, 13mpt2ex 7413 . . . 4 (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))) ∈ V
16 ccoid 16277 . . . . 5 comp = Slot (comp‘ndx)
1716setsid 16114 . . . 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)))⟩)))
1811, 15, 17mp2an 710 . . 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)))⟩))
1910, 18syl6eqr 2810 . 2 (𝜑 → (comp‘𝑂) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))))
20 simprr 813 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
21 simprl 811 . . . . . . 7 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑢 = ⟨𝑋, 𝑌⟩)
2221fveq2d 6354 . . . . . 6 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑢) = (2nd ‘⟨𝑋, 𝑌⟩))
231adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑋𝐵)
24 oppcco.y . . . . . . . 8 (𝜑𝑌𝐵)
2524adantr 472 . . . . . . 7 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑌𝐵)
26 op2ndg 7344 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2723, 25, 26syl2anc 696 . . . . . 6 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2822, 27eqtrd 2792 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑢) = 𝑌)
2920, 28opeq12d 4559 . . . 4 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ⟨𝑧, (2nd𝑢)⟩ = ⟨𝑍, 𝑌⟩)
3021fveq2d 6354 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑢) = (1st ‘⟨𝑋, 𝑌⟩))
31 op1stg 7343 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3223, 25, 31syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
3330, 32eqtrd 2792 . . . 4 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑢) = 𝑋)
3429, 33oveq12d 6829 . . 3 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)) = (⟨𝑍, 𝑌· 𝑋))
3534tposeqd 7522 . 2 ((𝜑 ∧ (𝑢 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)) = tpos (⟨𝑍, 𝑌· 𝑋))
36 opelxpi 5303 . . 3 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
371, 24, 36syl2anc 696 . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
38 oppcco.z . 2 (𝜑𝑍𝐵)
39 ovex 6839 . . . 4 (⟨𝑍, 𝑌· 𝑋) ∈ V
4039tposex 7553 . . 3 tpos (⟨𝑍, 𝑌· 𝑋) ∈ V
4140a1i 11 . 2 (𝜑 → tpos (⟨𝑍, 𝑌· 𝑋) ∈ V)
4219, 35, 37, 38, 41ovmpt2d 6951 1 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝑂)𝑍) = tpos (⟨𝑍, 𝑌· 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1630  wcel 2137  Vcvv 3338  cop 4325   × cxp 5262  cfv 6047  (class class class)co 6811  cmpt2 6813  1st c1st 7329  2nd c2nd 7330  tpos ctpos 7518  ndxcnx 16054   sSet csts 16055  Basecbs 16057  Hom chom 16152  compcco 16153  oppCatcoppc 16570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-cnex 10182  ax-resscn 10183  ax-1cn 10184  ax-icn 10185  ax-addcl 10186  ax-addrcl 10187  ax-mulcl 10188  ax-mulrcl 10189  ax-mulcom 10190  ax-addass 10191  ax-mulass 10192  ax-distr 10193  ax-i2m1 10194  ax-1ne0 10195  ax-1rid 10196  ax-rnegex 10197  ax-rrecex 10198  ax-cnre 10199  ax-pre-lttri 10200  ax-pre-lttrn 10201  ax-pre-ltadd 10202
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-tpos 7519  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-pnf 10266  df-mnf 10267  df-ltxr 10269  df-nn 11211  df-2 11269  df-3 11270  df-4 11271  df-5 11272  df-6 11273  df-7 11274  df-8 11275  df-9 11276  df-n0 11483  df-dec 11684  df-ndx 16060  df-slot 16061  df-sets 16064  df-cco 16167  df-oppc 16571
This theorem is referenced by:  oppcco  16576
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