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Theorem ringcvalALTV 48133
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 48132 . . . 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 3482 . . . . . 6 𝑢 ∈ V
54inex1 5323 . . . . 5 (𝑢 ∩ Ring) ∈ V
65a1i 11 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) ∈ V)
7 ineq1 4221 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
87adantl 481 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
9 ringcvalALTV.b . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Ring))
109adantr 480 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝐵 = (𝑈 ∩ Ring))
118, 10eqtr4d 2778 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) = 𝐵)
12 simpr 484 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
1312opeq2d 4885 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
14 eqidd 2736 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 RingHom 𝑦) = (𝑥 RingHom 𝑦))
1512, 12, 14mpoeq123dv 7508 . . . . . . 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 726 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
1815, 17eqtr4d 2778 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦)) = 𝐻)
1918opeq2d 4885 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
2012sqxpeqd 5721 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21 eqidd 2736 . . . . . . . 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 7508 . . . . . . 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 726 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
2522, 24eqtr4d 2778 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))) = · )
2625opeq2d 4885 . . . . 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 4753 . . . 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 3948 . . 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 3499 . . . 4 (𝑈𝑉𝑈 ∈ V)
3129, 30syl 17 . . 3 (𝜑𝑈 ∈ V)
32 tpex 7765 . . . 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 7023 . 2 (𝜑 → (RingCatALTV‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
351, 34eqtrid 2787 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  csb 3908  cin 3962  {ctp 4635  cop 4637  cmpt 5231   × cxp 5687  ccom 5693  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  ndxcnx 17227  Basecbs 17245  Hom chom 17309  compcco 17310  Ringcrg 20251   RingHom crh 20486  RingCatALTVcringcALTV 48131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-oprab 7435  df-mpo 7436  df-ringcALTV 48132
This theorem is referenced by:  ringcbasALTV  48144  ringchomfvalALTV  48145  ringccofvalALTV  48148
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