Theorem ringcval 42536

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	⊢ 𝐶 = (RingCat‘𝑈)
ringcval.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
ringcval.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
ringcval.h	⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
ringcval	⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Proof of Theorem ringcval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringcval.c	. 2 ⊢ 𝐶 = (RingCat‘𝑈)
2		df-ringc 42533	. . . 4 ⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
3	2	a1i 11	. . 3 ⊢ (𝜑 → RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))))
4		fveq2 6332	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5	4	adantl 467	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
6		ineq1 3958	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
7	6	sqxpeqd 5281	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
8		ringcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
9	8	sqxpeqd 5281	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
10	9	eqcomd 2777	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
11	7, 10	sylan9eqr 2827	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
12	11	reseq2d 5534	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
13		ringcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
14	13	eqcomd 2777	. . . . . 6 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
15	14	adantr 466	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
16	12, 15	eqtrd 2805	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
17	5, 16	oveq12d 6811	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
18		ringcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
19		elex 3364	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
20	18, 19	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
21		ovexd 6825	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
22	3, 17, 20, 21	fvmptd 6430	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
23	1, 22	syl5eq 2817	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ∩ cin 3722   ↦ cmpt 4863   × cxp 5247   ↾ cres 5251  ‘cfv 6031  (class class class)co 6793   ↾_cat cresc 16675  ExtStrCatcestrc 16969  Ringcrg 18755   RingHom crh 18922  RingCatcringc 42531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-ringc 42533

This theorem is referenced by:  ringcbas  42539  ringchomfval  42540  ringccofval  42544  dfringc2  42546  ringccat  42552  ringcid  42553  funcringcsetc  42563