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Theorem ringcval 46994
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 46991 . . 3 RingCat = (𝑒 ∈ V ↦ ((ExtStrCatβ€˜π‘’) β†Ύcat ( RingHom β†Ύ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring)))))
3 fveq2 6890 . . . . 5 (𝑒 = π‘ˆ β†’ (ExtStrCatβ€˜π‘’) = (ExtStrCatβ€˜π‘ˆ))
43adantl 480 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (ExtStrCatβ€˜π‘’) = (ExtStrCatβ€˜π‘ˆ))
5 ineq1 4204 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (𝑒 ∩ Ring) = (π‘ˆ ∩ Ring))
65sqxpeqd 5707 . . . . . . 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 5707 . . . . . . . 8 (πœ‘ β†’ (𝐡 Γ— 𝐡) = ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)))
98eqcomd 2736 . . . . . . 7 (πœ‘ β†’ ((π‘ˆ ∩ Ring) Γ— (π‘ˆ ∩ Ring)) = (𝐡 Γ— 𝐡))
106, 9sylan9eqr 2792 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring)) = (𝐡 Γ— 𝐡))
1110reseq2d 5980 . . . . 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 2736 . . . . . 6 (πœ‘ β†’ ( RingHom β†Ύ (𝐡 Γ— 𝐡)) = 𝐻)
1413adantr 479 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RingHom β†Ύ (𝐡 Γ— 𝐡)) = 𝐻)
1511, 14eqtrd 2770 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RingHom β†Ύ ((𝑒 ∩ Ring) Γ— (𝑒 ∩ Ring))) = 𝐻)
164, 15oveq12d 7429 . . 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 3493 . . 3 (πœ‘ β†’ π‘ˆ ∈ V)
19 ovexd 7446 . . 3 (πœ‘ β†’ ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 7005 . 2 (πœ‘ β†’ (RingCatβ€˜π‘ˆ) = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
211, 20eqtrid 2782 1 (πœ‘ β†’ 𝐢 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946   Γ— cxp 5673   β†Ύ cres 5677  β€˜cfv 6542  (class class class)co 7411   β†Ύcat cresc 17759  ExtStrCatcestrc 18077  Ringcrg 20127   RingHom crh 20360  RingCatcringc 46989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-ringc 46991
This theorem is referenced by:  ringcbas  46997  ringchomfval  46998  ringccofval  47002  dfringc2  47004  ringccat  47010  ringcid  47011  funcringcsetc  47021
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