Theorem rhmsscrnghm 20550

Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the non-unital ring homomorphisms between non-unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)

Hypotheses

Ref	Expression
rhmsscrnghm.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rhmsscrnghm.r	⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))
rhmsscrnghm.s	⊢ (𝜑 → 𝑆 = (Rng ∩ 𝑈))

Assertion

Ref	Expression
rhmsscrnghm	⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ (𝑆 × 𝑆)))

Proof of Theorem rhmsscrnghm

Dummy variables ℎ 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringrng 20170	. . . . . 6 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3	2	ssrdv 3941	. . . 4 ⊢ (𝜑 → Ring ⊆ Rng)
4	3	ssrind 4195	. . 3 ⊢ (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
5		rhmsscrnghm.r	. . 3 ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))
6		rhmsscrnghm.s	. . 3 ⊢ (𝜑 → 𝑆 = (Rng ∩ 𝑈))
7	4, 5, 6	3sstr4d 3991	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
8		ovres 7515	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
9	8	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
10	9	eleq2d 2814	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ℎ ∈ (𝑥 RingHom 𝑦)))
11		rhmisrnghm 20365	. . . . . 6 ⊢ (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥 RngHom 𝑦))
12	7	sseld 3934	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ 𝑆))
13	7	sseld 3934	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑅 → 𝑦 ∈ 𝑆))
14	12, 13	anim12d 609	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)))
15	14	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
16		ovres 7515	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHom 𝑦))
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHom 𝑦))
18	17	eleq2d 2814	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦) ↔ ℎ ∈ (𝑥 RngHom 𝑦)))
19	11, 18	imbitrrid 246	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦)))
20	10, 19	sylbid 240	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦)))
21	20	ssrdv 3941	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
22	21	ralrimivva 3172	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
23		inss1 4188	. . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring
24	5, 23	eqsstrdi 3980	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring)
25		xpss12 5634	. . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
26	24, 24, 25	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27		rhmfn 20384	. . . . 5 ⊢ RingHom Fn (Ring × Ring)
28		fnssresb 6604	. . . . 5 ⊢ ( RingHom Fn (Ring × Ring) → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
29	27, 28	mp1i 13	. . . 4 ⊢ (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
30	26, 29	mpbird 257	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
31		inss1 4188	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
32	6, 31	eqsstrdi 3980	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ Rng)
33		xpss12 5634	. . . . 5 ⊢ ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
34	32, 32, 33	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35		rnghmfn 20324	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
36		fnssresb 6604	. . . . 5 ⊢ ( RngHom Fn (Rng × Rng) → (( RngHom ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
37	35, 36	mp1i 13	. . . 4 ⊢ (𝜑 → (( RngHom ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
38	34, 37	mpbird 257	. . 3 ⊢ (𝜑 → ( RngHom ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
39		rhmsscrnghm.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
40		incom 4160	. . . . . 6 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41		inex1g 5258	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
42	40, 41	eqeltrid 2832	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (Rng ∩ 𝑈) ∈ V)
43	39, 42	syl 17	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V)
44	6, 43	eqeltrd 2828	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
45	30, 38, 44	isssc 17727	. 2 ⊢ (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ (𝑆 × 𝑆)) ↔ (𝑅 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))))
46	7, 22, 45	mpbir2and 713	1 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ (𝑆 × 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903   class class class wbr 5092   × cxp 5617   ↾ cres 5621   Fn wfn 6477  (class class class)co 7349   ⊆_cat cssc 17714  Rngcrng 20037  Ringcrg 20118   RngHom crnghm 20319   RingHom crh 20354

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-0g 17345  df-ssc 17717  df-mgm 18514  df-mgmhm 18566  df-sgrp 18593  df-mnd 18609  df-mhm 18657  df-grp 18815  df-minusg 18816  df-ghm 19092  df-cmn 19661  df-abl 19662  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-rnghm 20321  df-rhm 20357

This theorem is referenced by:  rhmsubcrngc  20553  rhmsubc  20574  rhmsubcALTV  48289