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Theorem rngcvalALTV 41246
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 41245 . . . 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 3189 . . . . . 6 𝑢 ∈ V
54inex1 4759 . . . . 5 (𝑢 ∩ Rng) ∈ V
65a1i 11 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Rng) ∈ V)
7 ineq1 3785 . . . . . 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 2658 . . . 4 ((𝜑𝑢 = 𝑈) → (𝑢 ∩ Rng) = 𝐵)
12 simpr 477 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
1312opeq2d 4377 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
14 eqidd 2622 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 RngHomo 𝑦) = (𝑥 RngHomo 𝑦))
1512, 12, 14mpt2eq123dv 6670 . . . . . . 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 761 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHomo 𝑦)))
1815, 17eqtr4d 2658 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦)) = 𝐻)
1918opeq2d 4377 . . . . 5 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHomo 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
2012sqxpeqd 5101 . . . . . . . 8 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21 eqidd 2622 . . . . . . . 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, 21mpt2eq123dv 6670 . . . . . . 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 761 . . . . . . 7 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))
2522, 24eqtr4d 2658 . . . . . 6 (((𝜑𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))) = · )
2625opeq2d 4377 . . . . 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 4253 . . . 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 3542 . . 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 3198 . . . 4 (𝑈𝑉𝑈 ∈ V)
3129, 30syl 17 . . 3 (𝜑𝑈 ∈ V)
32 tpex 6910 . . . 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 6245 . 2 (𝜑 → (RngCatALTV‘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 3514  cin 3554  {ctp 4152  cop 4154  cmpt 4673   × cxp 5072  ccom 5078  cfv 5847  (class class class)co 6604  cmpt2 6606  1st c1st 7111  2nd c2nd 7112  ndxcnx 15778  Basecbs 15781  Hom chom 15873  compcco 15874  Rngcrng 41159   RngHomo crngh 41170  RngCatALTVcrngcALTV 41243
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 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
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 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-oprab 6608  df-mpt2 6609  df-rngcALTV 41245
This theorem is referenced by:  rngcbasALTV  41268  rngchomfvalALTV  41269  rngccofvalALTV  41272
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