Theorem rngcval 45408

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 45405	. . 3 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
3		fveq2 6756	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
4	3	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5		ineq1 4136	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
6	5	sqxpeqd 5612	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
7		rngcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
8	7	sqxpeqd 5612	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
9	8	eqcomd 2744	. . . . . . 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 5880	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵)))
12		rngcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
13	12	eqcomd 2744	. . . . . 6 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
15	11, 14	eqtrd 2778	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
16	4, 15	oveq12d 7273	. . 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 3442	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
19		ovexd 7290	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
20	2, 16, 18, 19	fvmptd2 6865	. 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
21	1, 20	syl5eq 2791	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   × cxp 5578   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255   ↾_cat cresc 17437  ExtStrCatcestrc 17754  Rngcrng 45320   RngHomo crngh 45331  RngCatcrngc 45403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-rngc 45405

This theorem is referenced by:  rngcbas  45411  rngchomfval  45412  rngccofval  45416  dfrngc2  45418  rngccat  45424  rngcid  45425  rngcifuestrc  45443  funcrngcsetc  45444