Theorem rngcval 45531

Description: Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)

Hypotheses

Ref	Expression
rngcval.c	⊢ 𝐶 = (RngCat‘𝑈)
rngcval.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngcval.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
rngcval.h	⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))

Assertion

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

Proof of Theorem rngcval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngcval.c	. 2 ⊢ 𝐶 = (RngCat‘𝑈)
2		df-rngc 45528	. . 3 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
3		fveq2 6783	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
4	3	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5		ineq1 4140	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
6	5	sqxpeqd 5622	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
7		rngcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
8	7	sqxpeqd 5622	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
9	8	eqcomd 2745	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
10	6, 9	sylan9eqr 2801	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
11	10	reseq2d 5894	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵)))
12		rngcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
13	12	eqcomd 2745	. . . . . 6 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
14	13	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
15	11, 14	eqtrd 2779	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
16	4, 15	oveq12d 7302	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17		rngcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
18	17	elexd 3453	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
19		ovexd 7319	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
20	2, 16, 18, 19	fvmptd2 6892	. 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
21	1, 20	eqtrid 2791	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2107  Vcvv 3433   ∩ cin 3887   × cxp 5588   ↾ cres 5592  ‘cfv 6437  (class class class)co 7284   ↾_cat cresc 17529  ExtStrCatcestrc 17847  Rngcrng 45443   RngHomo crngh 45454  RngCatcrngc 45526

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 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-res 5602  df-iota 6395  df-fun 6439  df-fv 6445  df-ov 7287  df-rngc 45528

This theorem is referenced by:  rngcbas  45534  rngchomfval  45535  rngccofval  45539  dfrngc2  45541  rngccat  45547  rngcid  45548  rngcifuestrc  45566  funcrngcsetc  45567