rngcvalALTV - Mathbox for Alexander van der Vekens

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Theorem rngcvalALTV 47240

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 RngHom 𝑦)))
rngcvalALTV.o	⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘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 47239	. . . 4 ⊢ RngCatALTV = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Rng) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘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 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩}))
4		vex 3473	. . . . . 6 ⊢ 𝑢 ∈ V
5	4	inex1 5311	. . . . 5 ⊢ (𝑢 ∩ Rng) ∈ V
6	5	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Rng) ∈ V)
7		ineq1 4201	. . . . . 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 2770	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Rng) = 𝐵)
12		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
13	12	opeq2d 4876	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
14		eqidd 2728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 RngHom 𝑦) = (𝑥 RngHom 𝑦))
15	12, 12, 14	mpoeq123dv 7489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
16		rngcvalALTV.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
17	16	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
18	15, 17	eqtr4d 2770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦)) = 𝐻)
19	18	opeq2d 4876	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
20	12	sqxpeqd 5704	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
21		eqidd 2728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
22	20, 12, 21	mpoeq123dv 7489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
23		rngcvalALTV.o	. . . . . . . 8 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
24	23	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
25	22, 24	eqtr4d 2770	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = · )
26	25	opeq2d 4876	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
27	13, 19, 26	tpeq123d 4748	. . . 4 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
28	6, 11, 27	csbied2 3929	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⦋(𝑢 ∩ Rng) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2^nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1^st ‘𝑣) RngHom (2^nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
29		rngcvalALTV.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
30		elex 3488	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
31	29, 30	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
32		tpex 7741	. . . 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 7006	. 2 ⊢ (𝜑 → (RngCatALTV‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
35	1, 34	eqtrid 2779	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ⦋csb 3889 ∩ cin 3943 {ctp 4628 ⟨cop 4630 ↦ cmpt 5225 × cxp 5670 ∘ ccom 5676 ‘cfv 6542 (class class class)co 7414 ∈ cmpo 7416 1^st c1st 7983 2^nd c2nd 7984 ndxcnx 17147 Basecbs 17165 Hom chom 17229 compcco 17230 Rngcrng 20076 RngHom crnghm 20355 RngCatALTVcrngcALTV 47238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7732

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-oprab 7418 df-mpo 7419 df-rngcALTV 47239

This theorem is referenced by: rngcbasALTV 47241 rngchomfvalALTV 47242 rngccofvalALTV 47245

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