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Theorem ringccoALTV 42613
Description: Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringcbasALTV.c 𝐶 = (RingCatALTV‘𝑈)
ringcbasALTV.b 𝐵 = (Base‘𝐶)
ringcbasALTV.u (𝜑𝑈𝑉)
ringccoALTV.o · = (comp‘𝐶)
ringccoALTV.x (𝜑𝑋𝐵)
ringccoALTV.y (𝜑𝑌𝐵)
ringccoALTV.z (𝜑𝑍𝐵)
ringccoALTV.f (𝜑𝐹 ∈ (𝑋 RingHom 𝑌))
ringccoALTV.g (𝜑𝐺 ∈ (𝑌 RingHom 𝑍))
Assertion
Ref Expression
ringccoALTV (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Proof of Theorem ringccoALTV
Dummy variables 𝑓 𝑔 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringcbasALTV.c . . . 4 𝐶 = (RingCatALTV‘𝑈)
2 ringcbasALTV.b . . . 4 𝐵 = (Base‘𝐶)
3 ringcbasALTV.u . . . 4 (𝜑𝑈𝑉)
4 ringccoALTV.o . . . 4 · = (comp‘𝐶)
51, 2, 3, 4ringccofvalALTV 42612 . . 3 (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
6 simprl 778 . . . . . . 7 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑣 = ⟨𝑋, 𝑌⟩)
76fveq2d 6406 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = (2nd ‘⟨𝑋, 𝑌⟩))
8 ringccoALTV.x . . . . . . . 8 (𝜑𝑋𝐵)
9 ringccoALTV.y . . . . . . . 8 (𝜑𝑌𝐵)
10 op2ndg 7405 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
118, 9, 10syl2anc 575 . . . . . . 7 (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
1211adantr 468 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
137, 12eqtrd 2836 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (2nd𝑣) = 𝑌)
14 simprr 780 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → 𝑧 = 𝑍)
1513, 14oveq12d 6886 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((2nd𝑣) RingHom 𝑧) = (𝑌 RingHom 𝑍))
166fveq2d 6406 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = (1st ‘⟨𝑋, 𝑌⟩))
17 op1stg 7404 . . . . . . . 8 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
188, 9, 17syl2anc 575 . . . . . . 7 (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
1918adantr 468 . . . . . 6 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2016, 19eqtrd 2836 . . . . 5 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (1st𝑣) = 𝑋)
2120, 13oveq12d 6886 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → ((1st𝑣) RingHom (2nd𝑣)) = (𝑋 RingHom 𝑌))
22 eqidd 2803 . . . 4 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔𝑓) = (𝑔𝑓))
2315, 21, 22mpt2eq123dv 6941 . . 3 ((𝜑 ∧ (𝑣 = ⟨𝑋, 𝑌⟩ ∧ 𝑧 = 𝑍)) → (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔𝑓)))
24 opelxpi 5342 . . . 4 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
258, 9, 24syl2anc 575 . . 3 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
26 ringccoALTV.z . . 3 (𝜑𝑍𝐵)
27 ovex 6900 . . . . 5 (𝑌 RingHom 𝑍) ∈ V
28 ovex 6900 . . . . 5 (𝑋 RingHom 𝑌) ∈ V
2927, 28mpt2ex 7474 . . . 4 (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔𝑓)) ∈ V
3029a1i 11 . . 3 (𝜑 → (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔𝑓)) ∈ V)
315, 23, 25, 26, 30ovmpt2d 7012 . 2 (𝜑 → (⟨𝑋, 𝑌· 𝑍) = (𝑔 ∈ (𝑌 RingHom 𝑍), 𝑓 ∈ (𝑋 RingHom 𝑌) ↦ (𝑔𝑓)))
32 simprl 778 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑔 = 𝐺)
33 simprr 780 . . 3 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → 𝑓 = 𝐹)
3432, 33coeq12d 5482 . 2 ((𝜑 ∧ (𝑔 = 𝐺𝑓 = 𝐹)) → (𝑔𝑓) = (𝐺𝐹))
35 ringccoALTV.g . 2 (𝜑𝐺 ∈ (𝑌 RingHom 𝑍))
36 ringccoALTV.f . 2 (𝜑𝐹 ∈ (𝑋 RingHom 𝑌))
37 coexg 7341 . . 3 ((𝐺 ∈ (𝑌 RingHom 𝑍) ∧ 𝐹 ∈ (𝑋 RingHom 𝑌)) → (𝐺𝐹) ∈ V)
3835, 36, 37syl2anc 575 . 2 (𝜑 → (𝐺𝐹) ∈ V)
3931, 34, 35, 36, 38ovmpt2d 7012 1 (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2155  Vcvv 3387  cop 4370   × cxp 5303  ccom 5309  cfv 6095  (class class class)co 6868  cmpt2 6870  1st c1st 7390  2nd c2nd 7391  Basecbs 16062  compcco 16159   RingHom crh 18910  RingCatALTVcringcALTV 42566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173  ax-cnex 10271  ax-resscn 10272  ax-1cn 10273  ax-icn 10274  ax-addcl 10275  ax-addrcl 10276  ax-mulcl 10277  ax-mulrcl 10278  ax-mulcom 10279  ax-addass 10280  ax-mulass 10281  ax-distr 10282  ax-i2m1 10283  ax-1ne0 10284  ax-1rid 10285  ax-rnegex 10286  ax-rrecex 10287  ax-cnre 10288  ax-pre-lttri 10289  ax-pre-lttrn 10290  ax-pre-ltadd 10291  ax-pre-mulgt0 10292
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-nel 3078  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-int 4663  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-riota 6829  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-1st 7392  df-2nd 7393  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-1o 7790  df-oadd 7794  df-er 7973  df-en 8187  df-dom 8188  df-sdom 8189  df-fin 8190  df-pnf 10355  df-mnf 10356  df-xr 10357  df-ltxr 10358  df-le 10359  df-sub 10547  df-neg 10548  df-nn 11300  df-2 11358  df-3 11359  df-4 11360  df-5 11361  df-6 11362  df-7 11363  df-8 11364  df-9 11365  df-n0 11554  df-z 11638  df-dec 11754  df-uz 11899  df-fz 12544  df-struct 16064  df-ndx 16065  df-slot 16066  df-base 16068  df-hom 16171  df-cco 16172  df-ringcALTV 42568
This theorem is referenced by:  ringccatidALTV  42614  ringcsectALTV  42617  funcringcsetclem9ALTV  42629
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