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Theorem ringcval 46906
Description: Value of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
Hypotheses
Ref Expression
ringcval.c 𝐢 = (RingCatβ€˜π‘ˆ)
ringcval.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
ringcval.b (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Ring))
ringcval.h (πœ‘ β†’ 𝐻 = ( RingHom β†Ύ (𝐡 Γ— 𝐡)))
Assertion
Ref Expression
ringcval (πœ‘ β†’ 𝐢 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
Proof of Theorem ringcval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 ringcval.c . 2 𝐢 = (RingCatβ€˜π‘ˆ)
2 df-ringc 46903 . . 3 RingCat = (𝑒 ∈ V ↦ ((ExtStrCatβ€˜π‘’) β†Ύcat ( RingHom β†Ύ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring)))))
3 fveq2 6892 . . . . 5 (𝑒 = π‘ˆ β†’ (ExtStrCatβ€˜π‘’) = (ExtStrCatβ€˜π‘ˆ))
43adantl 483 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (ExtStrCatβ€˜π‘’) = (ExtStrCatβ€˜π‘ˆ))
5 ineq1 4206 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (𝑒 ∩ Ring) = (π‘ˆ ∩ Ring))
65sqxpeqd 5709 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring)) = ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)))
7 ringcval.b . . . . . . . . 9 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Ring))
87sqxpeqd 5709 . . . . . . . 8 (πœ‘ β†’ (𝐡 Γ— 𝐡) = ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)))
98eqcomd 2739 . . . . . . 7 (πœ‘ β†’ ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)) = (𝐡 Γ— 𝐡))
106, 9sylan9eqr 2795 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring)) = (𝐡 Γ— 𝐡))
1110reseq2d 5982 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RingHom β†Ύ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring))) = ( RingHom β†Ύ (𝐡 Γ— 𝐡)))
12 ringcval.h . . . . . . 7 (πœ‘ β†’ 𝐻 = ( RingHom β†Ύ (𝐡 Γ— 𝐡)))
1312eqcomd 2739 . . . . . 6 (πœ‘ β†’ ( RingHom β†Ύ (𝐡 Γ— 𝐡)) = 𝐻)
1413adantr 482 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RingHom β†Ύ (𝐡 Γ— 𝐡)) = 𝐻)
1511, 14eqtrd 2773 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RingHom β†Ύ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring))) = 𝐻)
164, 15oveq12d 7427 . . 3 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((ExtStrCatβ€˜π‘’) β†Ύcat ( RingHom β†Ύ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring)))) = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
17 ringcval.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
1817elexd 3495 . . 3 (πœ‘ β†’ π‘ˆ ∈ V)
19 ovexd 7444 . . 3 (πœ‘ β†’ ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 7007 . 2 (πœ‘ β†’ (RingCatβ€˜π‘ˆ) = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
211, 20eqtrid 2785 1 (πœ‘ β†’ 𝐢 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ∩ cin 3948   Γ— cxp 5675   β†Ύ cres 5679  β€˜cfv 6544  (class class class)co 7409   β†Ύcat cresc 17755  ExtStrCatcestrc 18073  Ringcrg 20056   RingHom crh 20248  RingCatcringc 46901
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-ringc 46903
This theorem is referenced by:  ringcbas  46909  ringchomfval  46910  ringccofval  46914  dfringc2  46916  ringccat  46922  ringcid  46923  funcringcsetc  46933
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