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Theorem ringcvalALTV 41311
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 41310 . . . 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 3189 . . . . . 6 𝑢 ∈ V
54inex1 4761 . . . . 5 (𝑢 ∩ Ring) ∈ V
65a1i 11 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) ∈ V)
7 ineq1 3787 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
87adantl 482 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
9 ringcvalALTV.b . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Ring))
109adantr 481 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝐵 = (𝑈 ∩ Ring))
118, 10eqtr4d 2658 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Ring) = 𝐵)
12 simpr 477 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
1312opeq2d 4379 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
14 eqidd 2622 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 RingHom 𝑦) = (𝑥 RingHom 𝑦))
1512, 12, 14mpt2eq123dv 6673 . . . . . . 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 761 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
1815, 17eqtr4d 2658 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦)) = 𝐻)
1918opeq2d 4379 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
2012sqxpeqd 5103 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21 eqidd 2622 . . . . . . . 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, 21mpt2eq123dv 6673 . . . . . . 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 761 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
2522, 24eqtr4d 2658 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))) = · )
2625opeq2d 4379 . . . . 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 4255 . . . 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 3543 . . 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 3198 . . . 4 (𝑈𝑉𝑈 ∈ V)
3129, 30syl 17 . . 3 (𝜑𝑈 ∈ V)
32 tpex 6913 . . . 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 6247 . 2 (𝜑 → (RingCatALTV‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
351, 34syl5eq 2667 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  csb 3515  cin 3555  {ctp 4154  cop 4156  cmpt 4675   × cxp 5074  ccom 5080  cfv 5849  (class class class)co 6607  cmpt2 6609  1st c1st 7114  2nd c2nd 7115  ndxcnx 15781  Basecbs 15784  Hom chom 15876  compcco 15877  Ringcrg 18471   RingHom crh 18636  RingCatALTVcringcALTV 41308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fv 5857  df-oprab 6611  df-mpt2 6612  df-ringcALTV 41310
This theorem is referenced by:  ringcbasALTV  41350  ringchomfvalALTV  41351  ringccofvalALTV  41354
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