Theorem rhmsscrnghm 46877

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 ( RngHomo ↾ (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 46641	. . . . . 6 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3	2	ssrdv 3987	. . . 4 ⊢ (𝜑 → Ring ⊆ Rng)
4	3	ssrind 4234	. . 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 4028	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
8		ovres 7569	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
9	8	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
10	9	eleq2d 2819	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ℎ ∈ (𝑥 RingHom 𝑦)))
11		rhmisrnghm 46707	. . . . . 6 ⊢ (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥 RngHomo 𝑦))
12	7	sseld 3980	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ 𝑆))
13	7	sseld 3980	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑅 → 𝑦 ∈ 𝑆))
14	12, 13	anim12d 609	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)))
15	14	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
16		ovres 7569	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
18	17	eleq2d 2819	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) ↔ ℎ ∈ (𝑥 RngHomo 𝑦)))
19	11, 18	imbitrrid 245	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
20	10, 19	sylbid 239	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ℎ ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
21	20	ssrdv 3987	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
22	21	ralrimivva 3200	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
23		inss1 4227	. . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring
24	5, 23	eqsstrdi 4035	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring)
25		xpss12 5690	. . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
26	24, 24, 25	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27		rhmfn 46705	. . . . 5 ⊢ RingHom Fn (Ring × Ring)
28		fnssresb 6669	. . . . 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 256	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
31		inss1 4227	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
32	6, 31	eqsstrdi 4035	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ Rng)
33		xpss12 5690	. . . . 5 ⊢ ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
34	32, 32, 33	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35		rnghmfn 46673	. . . . 5 ⊢ RngHomo Fn (Rng × Rng)
36		fnssresb 6669	. . . . 5 ⊢ ( RngHomo Fn (Rng × Rng) → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
37	35, 36	mp1i 13	. . . 4 ⊢ (𝜑 → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
38	34, 37	mpbird 256	. . 3 ⊢ (𝜑 → ( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
39		rhmsscrnghm.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
40		incom 4200	. . . . . 6 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41		inex1g 5318	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
42	40, 41	eqeltrid 2837	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (Rng ∩ 𝑈) ∈ V)
43	39, 42	syl 17	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V)
44	6, 43	eqeltrd 2833	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
45	30, 38, 44	isssc 17763	. 2 ⊢ (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)) ↔ (𝑅 ⊆ 𝑆 ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))))
46	7, 22, 45	mpbir2and 711	1 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∩ cin 3946   ⊆ wss 3947   class class class wbr 5147   × cxp 5673   ↾ cres 5677   Fn wfn 6535  (class class class)co 7405   ⊆_cat cssc 17750  Ringcrg 20049   RingHom crh 20240  Rngcrng 46634   RngHomo crngh 46668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-0g 17383  df-ssc 17753  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-grp 18818  df-minusg 18819  df-ghm 19084  df-cmn 19644  df-abl 19645  df-mgp 19982  df-ur 19999  df-ring 20051  df-rnghom 20243  df-mgmhm 46535  df-rng 46635  df-rnghomo 46670

This theorem is referenced by:  rhmsubcrngc  46880  rhmsubc  46941  rhmsubcALTV  46959