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Theorem ringcvalALTV 47666
Description: Value of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringcvalALTV.c 𝐶 = (RingCatALTV‘𝑈)
ringcvalALTV.u (𝜑𝑈𝑉)
ringcvalALTV.b (𝜑𝐵 = (𝑈 ∩ Ring))
ringcvalALTV.h (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
ringcvalALTV.o (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
Assertion
Ref Expression
ringcvalALTV (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Distinct variable groups:   𝑓,𝑔,𝑣,𝑥,𝑦,𝑧   𝑣,𝐵,𝑥,𝑦,𝑧   𝑣,𝑈,𝑥,𝑦,𝑧   𝜑,𝑣,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   · (𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑈(𝑓,𝑔)   𝐻(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)

Proof of Theorem ringcvalALTV
Dummy variables 𝑏 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringcvalALTV.c . 2 𝐶 = (RingCatALTV‘𝑈)
2 df-ringcALTV 47665 . . . 4 RingCatALTV = (𝑢 ∈ V ↦ (𝑢 ∩ Ring) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)))⟩})
32a1i 11 . . 3 (𝜑 → RingCatALTV = (𝑢 ∈ V ↦ (𝑢 ∩ Ring) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)))⟩}))
4 vex 3466 . . . . . 6 𝑢 ∈ V
54inex1 5322 . . . . 5 (𝑢 ∩ Ring) ∈ V
65a1i 11 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) ∈ V)
7 ineq1 4206 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
87adantl 480 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
9 ringcvalALTV.b . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Ring))
109adantr 479 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝐵 = (𝑈 ∩ Ring))
118, 10eqtr4d 2769 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) = 𝐵)
12 simpr 483 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
1312opeq2d 4886 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
14 eqidd 2727 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 RingHom 𝑦) = (𝑥 RingHom 𝑦))
1512, 12, 14mpoeq123dv 7500 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
16 ringcvalALTV.h . . . . . . . 8 (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
1716ad2antrr 724 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
1815, 17eqtr4d 2769 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦)) = 𝐻)
1918opeq2d 4886 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
2012sqxpeqd 5714 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21 eqidd 2727 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)))
2220, 12, 21mpoeq123dv 7500 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
23 ringcvalALTV.o . . . . . . . 8 (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
2423ad2antrr 724 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
2522, 24eqtr4d 2769 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))) = · )
2625opeq2d 4886 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
2713, 19, 26tpeq123d 4757 . . . 4 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
286, 11, 27csbied2 3932 . . 3 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
29 ringcvalALTV.u . . . 4 (𝜑𝑈𝑉)
30 elex 3482 . . . 4 (𝑈𝑉𝑈 ∈ V)
3129, 30syl 17 . . 3 (𝜑𝑈 ∈ V)
32 tpex 7755 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
3332a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
343, 28, 31, 33fvmptd 7016 . 2 (𝜑 → (RingCatALTV‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
351, 34eqtrid 2778 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  csb 3892  cin 3946  {ctp 4637  cop 4639  cmpt 5236   × cxp 5680  ccom 5686  cfv 6554  (class class class)co 7424  cmpo 7426  1st c1st 8001  2nd c2nd 8002  ndxcnx 17195  Basecbs 17213  Hom chom 17277  compcco 17278  Ringcrg 20216   RingHom crh 20451  RingCatALTVcringcALTV 47664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6506  df-fun 6556  df-fv 6562  df-oprab 7428  df-mpo 7429  df-ringcALTV 47665
This theorem is referenced by:  ringcbasALTV  47677  ringchomfvalALTV  47678  ringccofvalALTV  47681
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