Theorem dfrngc2 20544

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 20534	. 2 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
6		eqid 2730	. . 3 ⊢ ((ExtStrCat‘𝑈) ↾cat 𝐻) = ((ExtStrCat‘𝑈) ↾cat 𝐻)
7		fvexd 6876	. . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) ∈ V)
8		inex1g 5277	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
9	2, 8	syl 17	. . . 4 ⊢ (𝜑 → (𝑈 ∩ Rng) ∈ V)
10	3, 9	eqeltrd 2829	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
11	3, 4	rnghmresfn 20535	. . 3 ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
12	6, 7, 10, 11	rescval2 17797	. 2 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) = (((ExtStrCat‘𝑈) ↾s 𝐵) sSet ⟨(Hom ‘ndx), 𝐻⟩))
13		eqid 2730	. . . 4 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
14		eqidd 2731	. . . 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 2730	. . . . . 6 ⊢ (comp‘(ExtStrCat‘𝑈)) = (comp‘(ExtStrCat‘𝑈))
17	13, 2, 16	estrccofval 18097	. . . . 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 2765	. . . 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 18092	. . 3 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), · ⟩})
20		mpoexga 8059	. . . 4 ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑈 ∈ 𝑉) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
21	2, 2, 20	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
22		fvexd 6876	. . . 4 ⊢ (𝜑 → (comp‘(ExtStrCat‘𝑈)) ∈ V)
23	15, 22	eqeltrd 2829	. . 3 ⊢ (𝜑 → · ∈ V)
24		rnghmfn 20355	. . . . . 6 ⊢ RngHom Fn (Rng × Rng)
25		fnfun 6621	. . . . . 6 ⊢ ( RngHom Fn (Rng × Rng) → Fun RngHom )
26	24, 25	mp1i 13	. . . . 5 ⊢ (𝜑 → Fun RngHom )
27		sqxpexg 7734	. . . . . 6 ⊢ (𝐵 ∈ V → (𝐵 × 𝐵) ∈ V)
28	10, 27	syl 17	. . . . 5 ⊢ (𝜑 → (𝐵 × 𝐵) ∈ V)
29		resfunexg 7192	. . . . 5 ⊢ ((Fun RngHom ∧ (𝐵 × 𝐵) ∈ V) → ( RngHom ↾ (𝐵 × 𝐵)) ∈ V)
30	26, 28, 29	syl2anc 584	. . . 4 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) ∈ V)
31	4, 30	eqeltrd 2829	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
32		inss1 4203	. . . 4 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
33	3, 32	eqsstrdi 3994	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
34	19, 2, 21, 23, 31, 33	estrres 18107	. 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), · ⟩})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∩ cin 3916  {ctp 4596  ⟨cop 4598   × cxp 5639   ↾ cres 5643   ∘ ccom 5645  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  1^st c1st 7969  2^nd c2nd 7970   ↑_m cmap 8802   sSet csts 17140  ndxcnx 17170  Basecbs 17186   ↾_s cress 17207  Hom chom 17238  compcco 17239   ↾_cat cresc 17777  ExtStrCatcestrc 18090  Rngcrng 20068   RngHom crnghm 20350  RngCatcrngc 20532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-hom 17251  df-cco 17252  df-resc 17780  df-estrc 18091  df-rnghm 20352  df-rngc 20533

This theorem is referenced by:  rngcresringcat  20585