Theorem rngcvalALTV 45471

Description: Value of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)

Hypotheses

Ref	Expression
rngcvalALTV.c	⊢ 𝐶 = (RngCatALTV‘𝑈)
rngcvalALTV.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngcvalALTV.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
rngcvalALTV.h	⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)))
rngcvalALTV.o	⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))

Assertion

Ref	Expression
rngcvalALTV	⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Distinct variable groups:   𝑓,𝑔,𝑣,𝑥,𝑦,𝑧   𝑣,𝐵,𝑥,𝑦,𝑧   𝑣,𝑈,𝑥,𝑦,𝑧   𝜑,𝑣,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   · (𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑈(𝑓,𝑔)   𝐻(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)

Proof of Theorem rngcvalALTV

Dummy variables 𝑏 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngcvalALTV.c	. 2 ⊢ 𝐶 = (RngCatALTV‘𝑈)
2		df-rngcALTV 45470	. . . 4 ⊢ RngCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Rng) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
3	2	a1i 11	. . 3 ⊢ (𝜑 → RngCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Rng) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩}))
4		vex 3434	. . . . . 6 ⊢ 𝑢 ∈ V
5	4	inex1 5244	. . . . 5 ⊢ (𝑢 ∩ Rng) ∈ V
6	5	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Rng) ∈ V)
7		ineq1 4144	. . . . . 6 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
8	7	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
9		rngcvalALTV.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
10	9	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝐵 = (𝑈 ∩ Rng))
11	8, 10	eqtr4d 2782	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Rng) = 𝐵)
12		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
13	12	opeq2d 4816	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
14		eqidd 2740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 RngHomo 𝑦) = (𝑥 RngHomo 𝑦))
15	12, 12, 14	mpoeq123dv 7341	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)))
16		rngcvalALTV.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)))
17	16	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦)))
18	15, 17	eqtr4d 2782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦)) = 𝐻)
19	18	opeq2d 4816	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
20	12	sqxpeqd 5620	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21		eqidd 2740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
22	20, 12, 21	mpoeq123dv 7341	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
23		rngcvalALTV.o	. . . . . . . 8 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
24	23	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
25	22, 24	eqtr4d 2782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = · )
26	25	opeq2d 4816	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
27	13, 19, 26	tpeq123d 4689	. . . 4 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
28	6, 11, 27	csbied2 3876	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⦋(𝑢 ∩ Rng) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
29		rngcvalALTV.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
30		elex 3448	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
31	29, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
32		tpex 7588	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
33	32	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
34	3, 28, 31, 33	fvmptd 6876	. 2 ⊢ (𝜑 → (RngCatALTV‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
35	1, 34	eqtrid 2791	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⦋csb 3836   ∩ cin 3890  {ctp 4570  ⟨cop 4572   ↦ cmpt 5161   × cxp 5586   ∘ ccom 5592  ‘cfv 6430  (class class class)co 7268   ∈ cmpo 7270  1^st c1st 7815  2^nd c2nd 7816  ndxcnx 16875  Basecbs 16893  Hom chom 16954  compcco 16955  Rngcrng 45384   RngHomo crngh 45395  RngCatALTVcrngcALTV 45468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  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 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-iota 6388  df-fun 6432  df-fv 6438  df-oprab 7272  df-mpo 7273  df-rngcALTV 45470

This theorem is referenced by:  rngcbasALTV  45493  rngchomfvalALTV  45494  rngccofvalALTV  45497