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Theorem rngcvalALTV 45519
Description: Value of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
Hypotheses
Ref Expression
rngcvalALTV.c 𝐶 = (RngCatALTV‘𝑈)
rngcvalALTV.u (𝜑𝑈𝑉)
rngcvalALTV.b (𝜑𝐵 = (𝑈 ∩ Rng))
rngcvalALTV.h (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
rngcvalALTV.o (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))
Assertion
Ref Expression
rngcvalALTV (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 rngcvalALTV
Dummy variables 𝑏 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngcvalALTV.c . 2 𝐶 = (RngCatALTV‘𝑈)
2 df-rngcALTV 45518 . . . 4 RngCatALTV = (𝑢 ∈ V ↦ (𝑢 ∩ Rng) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)))⟩})
32a1i 11 . . 3 (𝜑 → RngCatALTV = (𝑢 ∈ V ↦ (𝑢 ∩ Rng) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)))⟩}))
4 vex 3436 . . . . . 6 𝑢 ∈ V
54inex1 5241 . . . . 5 (𝑢 ∩ Rng) ∈ V
65a1i 11 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Rng) ∈ V)
7 ineq1 4139 . . . . . 6 (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
87adantl 482 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
9 rngcvalALTV.b . . . . . 6 (𝜑𝐵 = (𝑈 ∩ Rng))
109adantr 481 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝐵 = (𝑈 ∩ Rng))
118, 10eqtr4d 2781 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Rng) = 𝐵)
12 simpr 485 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
1312opeq2d 4811 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
14 eqidd 2739 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 RngHomo 𝑦) = (𝑥 RngHomo 𝑦))
1512, 12, 14mpoeq123dv 7350 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦)) = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
16 rngcvalALTV.h . . . . . . . 8 (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
1716ad2antrr 723 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
1815, 17eqtr4d 2781 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦)) = 𝐻)
1918opeq2d 4811 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
2012sqxpeqd 5621 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21 eqidd 2739 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)))
2220, 12, 21mpoeq123dv 7350 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))
23 rngcvalALTV.o . . . . . . . 8 (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))
2423ad2antrr 723 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))
2522, 24eqtr4d 2781 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))) = · )
2625opeq2d 4811 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
2713, 19, 26tpeq123d 4684 . . . 4 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
286, 11, 27csbied2 3872 . . 3 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Rng) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
29 rngcvalALTV.u . . . 4 (𝜑𝑈𝑉)
30 elex 3450 . . . 4 (𝑈𝑉𝑈 ∈ V)
3129, 30syl 17 . . 3 (𝜑𝑈 ∈ V)
32 tpex 7597 . . . 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 6882 . 2 (𝜑 → (RngCatALTV‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
351, 34eqtrid 2790 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  csb 3832  cin 3886  {ctp 4565  cop 4567  cmpt 5157   × cxp 5587  ccom 5593  cfv 6433  (class class class)co 7275  cmpo 7277  1st c1st 7829  2nd c2nd 7830  ndxcnx 16894  Basecbs 16912  Hom chom 16973  compcco 16974  Rngcrng 45432   RngHomo crngh 45443  RngCatALTVcrngcALTV 45516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-oprab 7279  df-mpo 7280  df-rngcALTV 45518
This theorem is referenced by:  rngcbasALTV  45541  rngchomfvalALTV  45542  rngccofvalALTV  45545
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