Theorem dfrngc2 42490

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 = ( RngHomo ↾ (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 = ( RngHomo ↾ (𝐵 × 𝐵)))
5	1, 2, 3, 4	rngcval 42480	. 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6		eqid 2770	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7		fvexd 6344	. . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V)
8		inex1g 4932	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
9	2, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
10	3, 9	eqeltrd 2849	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
11	3, 4	rnghmresfn 42481	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
12	6, 7, 10, 11	rescval2 16694	. 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
13		eqid 2770	. . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
14		eqidd 2771	. . . 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 2770	. . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
17	13, 2, 16	estrccofval 16975	. . . . 5 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
18	15, 17	eqtrd 2804	. . . 4 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
19	13, 2, 14, 18	estrcval 16970	. . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
20		mpt2exga 7395	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
21	2, 2, 20	syl2anc 565	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
22		fvexd 6344	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
23	15, 22	eqeltrd 2849	. . 3 ⊢ (𝜑 → · ∈ V)
24		rnghmfn 42408	. . . . . 6 ⊢ RngHomo Fn (Rng × Rng)
25		fnfun 6128	. . . . . 6 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
26	24, 25	mp1i 13	. . . . 5 ⊢ (𝜑 → Fun RngHomo )
27		sqxpexg 7109	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	10, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 6622	. . . . 5 ⊢ ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
30	26, 28, 29	syl2anc 565	. . . 4 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
31	4, 30	eqeltrd 2849	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		inss1 3979	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
33	3, 32	syl6eqss 3802	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
34	19, 2, 21, 23, 31, 33	estrres 16986	. 2 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
35	5, 12, 34	3eqtrd 2808	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ∩ cin 3720  {ctp 4318  ⟨cop 4320   × cxp 5247   ↾ cres 5251   ∘ ccom 5253  Fun wfun 6025   Fn wfn 6026  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794  1^st c1st 7312  2^nd c2nd 7313   ↑_𝑚 cmap 8008  ndxcnx 16060   sSet csts 16061  Basecbs 16063   ↾_s cress 16064  Hom chom 16159  compcco 16160   ↾_cat cresc 16674  ExtStrCatcestrc 16968  Rngcrng 42392   RngHomo crngh 42403  RngCatcrngc 42475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-hom 16173  df-cco 16174  df-resc 16677  df-estrc 16969  df-rnghomo 42405  df-rngc 42477

This theorem is referenced by:  rngcresringcat  42548