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Theorem dfrngc2 20565

Description: Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)

**Hypotheses**
Ref	Expression
dfrngc2.c	⊢ 𝐶 = (RngCat‘𝑈)
dfrngc2.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
dfrngc2.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
dfrngc2.h	⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
dfrngc2.o	⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))

**Assertion**
Ref	Expression
dfrngc2	⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Proof of Theorem dfrngc2 Dummy variables 𝑓 𝑔 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrngc2.c	. . 3 ⊢ 𝐶 = (RngCat‘𝑈)
2		dfrngc2.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
3		dfrngc2.b	. . 3 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
4		dfrngc2.h	. . 3 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
5	1, 2, 3, 4	rngcval 20555	. 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾_cat 𝐻))
6		eqid 2725	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾_cat 𝐻) = ((ExtStrCat‘𝑈) ↾_cat 𝐻)
7		fvexd 6907	. . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V)
8		inex1g 5314	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
9	2, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
10	3, 9	eqeltrd 2825	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
11	3, 4	rnghmresfn 20556	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
12	6, 7, 10, 11	rescval2 17810	. 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾_cat 𝐻) = (((ExtStrCat‘𝑈) ↾_s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
13		eqid 2725	. . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
14		eqidd 2726	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))))
15		dfrngc2.o	. . . . 5 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))
16		eqid 2725	. . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
17	13, 2, 16	estrccofval 18118	. . . . 5 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
18	15, 17	eqtrd 2765	. . . 4 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
19	13, 2, 14, 18	estrcval 18113	. . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
20		mpoexga 8080	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))) ∈ V)
21	2, 2, 20	syl2anc 582	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))) ∈ V)
22		fvexd 6907	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
23	15, 22	eqeltrd 2825	. . 3 ⊢ (𝜑 → · ∈ V)
24		rnghmfn 20382	. . . . . 6 ⊢ RngHom Fn (Rng × Rng)
25		fnfun 6649	. . . . . 6 ⊢ ( RngHom Fn (Rng × Rng) → Fun RngHom )
26	24, 25	mp1i 13	. . . . 5 ⊢ (𝜑 → Fun RngHom )
27		sqxpexg 7755	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	10, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 7223	. . . . 5 ⊢ ((Fun RngHom ∧ (𝐵 × 𝐵) ∈ V) → ( RngHom ↾ (𝐵 × 𝐵)) ∈ V)
30	26, 28, 29	syl2anc 582	. . . 4 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) ∈ V)
31	4, 30	eqeltrd 2825	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		inss1 4223	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
33	3, 32	eqsstrdi 4027	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
34	19, 2, 21, 23, 31, 33	estrres 18129	. 2 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾_s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
35	5, 12, 34	3eqtrd 2769	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ∩ cin 3938 {ctp 4628 ⟨cop 4630 × cxp 5670 ↾ cres 5674 ∘ ccom 5676 Fun wfun 6537 Fn wfn 6538 ‘cfv 6543 (class class class)co 7416 ∈ cmpo 7418 1^st c1st 7989 2^nd c2nd 7990 ↑_m cmap 8843 sSet csts 17131 ndxcnx 17161 Basecbs 17179 ↾_s cress 17208 Hom chom 17243 compcco 17244 ↾_cat cresc 17790 ExtStrCatcestrc 18111 Rngcrng 20096 RngHom crnghm 20377 RngCatcrngc 20553

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-hom 17256 df-cco 17257 df-resc 17793 df-estrc 18112 df-rnghm 20379 df-rngc 20554

This theorem is referenced by: rngcresringcat 20606

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