Theorem dfrngc2 41764

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 41754	. 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6		eqid 2610	. . . 4 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7		fvex 6113	. . . . 5 ⊢ (ExtStrCat‘𝑈) ∈ V
8	7	a1i 11	. . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V)
9		inex1g 4729	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
10	2, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
11	3, 10	eqeltrd 2688	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
12	3, 4	rnghmresfn 41755	. . . 4 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
13	6, 8, 11, 12	rescval2 16311	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
14		eqid 2610	. . . . 5 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
15		eqidd 2611	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
16		dfrngc2.o	. . . . . 6 ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))
17		eqid 2610	. . . . . . 7 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
18	14, 2, 17	estrccofval 16592	. . . . . 6 ⊢ (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))))
19	16, 18	eqtrd 2644	. . . . 5 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
20	14, 2, 15, 19	estrcval 16587	. . . 4 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
21	2, 2	jca 553	. . . . 5 ⊢ (𝜑 → (𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉))
22		eqid 2610	. . . . . 6 ⊢ (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
23	22	mpt2exg 7134	. . . . 5 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
24	21, 23	syl 17	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ∈ V)
25		fvex 6113	. . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) ∈ V
26	25	a1i 11	. . . . 5 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
27	16, 26	eqeltrd 2688	. . . 4 ⊢ (𝜑 → · ∈ V)
28		rnghmfn 41680	. . . . . . 7 ⊢ RngHomo Fn (Rng × Rng)
29		fnfun 5902	. . . . . . 7 ⊢ ( RngHomo Fn (Rng × Rng) → Fun RngHomo )
30	28, 29	mp1i 13	. . . . . 6 ⊢ (𝜑 → Fun RngHomo )
31		sqxpexg 6861	. . . . . . 7 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
32	11, 31	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
33		resfunexg 6384	. . . . . 6 ⊢ ((Fun RngHomo ∧ (𝐵 × 𝐵) ∈ V) → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
34	30, 32, 33	syl2anc 691	. . . . 5 ⊢ (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) ∈ V)
35	4, 34	eqeltrd 2688	. . . 4 ⊢ (𝜑 → 𝐻 ∈ V)
36		inss1 3795	. . . . . 6 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈)
38	3	sseq1d 3595	. . . . 5 ⊢ (𝜑 → (𝐵 ⊆ 𝑈 ↔ (𝑈 ∩ Rng) ⊆ 𝑈))
39	37, 38	mpbird 246	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
40	20, 2, 24, 27, 11, 35, 39	estrres 16602	. . 3 ⊢ (𝜑 → (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
41	13, 40	eqtrd 2644	. 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
42	5, 41	eqtrd 2644	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  {ctp 4129  ⟨cop 4131   × cxp 5036   ↾ cres 5040   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744  ndxcnx 15692   sSet csts 15693  Basecbs 15695   ↾_s cress 15696  Hom chom 15779  compcco 15780   ↾_cat cresc 16291  ExtStrCatcestrc 16585  Rngcrng 41664   RngHomo crngh 41675  RngCatcrngc 41749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-hom 15793  df-cco 15794  df-resc 16294  df-estrc 16586  df-rnghomo 41677  df-rngc 41751

This theorem is referenced by:  rngcresringcat  41822