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Theorem rngcvalALTV 47435
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 𝐢 = (RngCatALTVβ€˜π‘ˆ)
rngcvalALTV.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngcvalALTV.b (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Rng))
rngcvalALTV.h (πœ‘ β†’ 𝐻 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (π‘₯ RngHom 𝑦)))
rngcvalALTV.o (πœ‘ β†’ Β· = (𝑣 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))))
Assertion
Ref Expression
rngcvalALTV (πœ‘ β†’ 𝐢 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
Distinct variable groups:   𝑓,𝑔,𝑣,π‘₯,𝑦,𝑧   𝑣,𝐡,π‘₯,𝑦,𝑧   𝑣,π‘ˆ,π‘₯,𝑦,𝑧   πœ‘,𝑣,π‘₯,𝑦,𝑧
Allowed substitution hints:   πœ‘(𝑓,𝑔)   𝐡(𝑓,𝑔)   𝐢(π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)   Β· (π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)   π‘ˆ(𝑓,𝑔)   𝐻(π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑉(π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)

Proof of Theorem rngcvalALTV
Dummy variables 𝑏 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rngcvalALTV.c . 2 𝐢 = (RngCatALTVβ€˜π‘ˆ)
2 df-rngcALTV 47434 . . . 4 RngCatALTV = (𝑒 ∈ V ↦ ⦋(𝑒 ∩ Rng) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ RngHom 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩})
32a1i 11 . . 3 (πœ‘ β†’ RngCatALTV = (𝑒 ∈ V ↦ ⦋(𝑒 ∩ Rng) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ RngHom 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩}))
4 vex 3467 . . . . . 6 𝑒 ∈ V
54inex1 5313 . . . . 5 (𝑒 ∩ Rng) ∈ V
65a1i 11 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑒 ∩ Rng) ∈ V)
7 ineq1 4200 . . . . . 6 (𝑒 = π‘ˆ β†’ (𝑒 ∩ Rng) = (π‘ˆ ∩ Rng))
87adantl 480 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑒 ∩ Rng) = (π‘ˆ ∩ Rng))
9 rngcvalALTV.b . . . . . 6 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Rng))
109adantr 479 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ 𝐡 = (π‘ˆ ∩ Rng))
118, 10eqtr4d 2768 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑒 ∩ Rng) = 𝐡)
12 simpr 483 . . . . . 6 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ 𝑏 = 𝐡)
1312opeq2d 4877 . . . . 5 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ ⟨(Baseβ€˜ndx), π‘βŸ© = ⟨(Baseβ€˜ndx), 𝐡⟩)
14 eqidd 2726 . . . . . . . 8 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ (π‘₯ RngHom 𝑦) = (π‘₯ RngHom 𝑦))
1512, 12, 14mpoeq123dv 7489 . . . . . . 7 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ RngHom 𝑦)) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (π‘₯ RngHom 𝑦)))
16 rngcvalALTV.h . . . . . . . 8 (πœ‘ β†’ 𝐻 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (π‘₯ RngHom 𝑦)))
1716ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ 𝐻 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ (π‘₯ RngHom 𝑦)))
1815, 17eqtr4d 2768 . . . . . 6 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ RngHom 𝑦)) = 𝐻)
1918opeq2d 4877 . . . . 5 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ RngHom 𝑦))⟩ = ⟨(Hom β€˜ndx), 𝐻⟩)
2012sqxpeqd 5705 . . . . . . . 8 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ (𝑏 Γ— 𝑏) = (𝐡 Γ— 𝐡))
21 eqidd 2726 . . . . . . . 8 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))
2220, 12, 21mpoeq123dv 7489 . . . . . . 7 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))))
23 rngcvalALTV.o . . . . . . . 8 (πœ‘ β†’ Β· = (𝑣 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))))
2423ad2antrr 724 . . . . . . 7 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ Β· = (𝑣 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))))
2522, 24eqtr4d 2768 . . . . . 6 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓))) = Β· )
2625opeq2d 4877 . . . . 5 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(compβ€˜ndx), Β· ⟩)
2713, 19, 26tpeq123d 4749 . . . 4 (((πœ‘ ∧ 𝑒 = π‘ˆ) ∧ 𝑏 = 𝐡) β†’ {⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ RngHom 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
286, 11, 27csbied2 3926 . . 3 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⦋(𝑒 ∩ Rng) / π‘β¦Œ{⟨(Baseβ€˜ndx), π‘βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (π‘₯ RngHom 𝑦))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑏 Γ— 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd β€˜π‘£) RngHom 𝑧), 𝑓 ∈ ((1st β€˜π‘£) RngHom (2nd β€˜π‘£)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
29 rngcvalALTV.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
30 elex 3482 . . . 4 (π‘ˆ ∈ 𝑉 β†’ π‘ˆ ∈ V)
3129, 30syl 17 . . 3 (πœ‘ β†’ π‘ˆ ∈ V)
32 tpex 7744 . . . 4 {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩} ∈ V
3332a1i 11 . . 3 (πœ‘ β†’ {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩} ∈ V)
343, 28, 31, 33fvmptd 7005 . 2 (πœ‘ β†’ (RngCatALTVβ€˜π‘ˆ) = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
351, 34eqtrid 2777 1 (πœ‘ β†’ 𝐢 = {⟨(Baseβ€˜ndx), 𝐡⟩, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463  β¦‹csb 3886   ∩ cin 3940  {ctp 4629  βŸ¨cop 4631   ↦ cmpt 5227   Γ— cxp 5671   ∘ ccom 5677  β€˜cfv 6543  (class class class)co 7413   ∈ cmpo 7415  1st c1st 7985  2nd c2nd 7986  ndxcnx 17156  Basecbs 17174  Hom chom 17238  compcco 17239  Rngcrng 20091   RngHom crnghm 20372  RngCatALTVcrngcALTV 47433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-oprab 7417  df-mpo 7418  df-rngcALTV 47434
This theorem is referenced by:  rngcbasALTV  47436  rngchomfvalALTV  47437  rngccofvalALTV  47440
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