Theorem rhmsscrnghm 47013

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 20174	. . . . . 6 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3	2	ssrdv 3988	. . . 4 ⊢ (𝜑 → Ring ⊆ Rng)
4	3	ssrind 4235	. . 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 4029	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
8		ovres 7577	. . . . . . 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 2818	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ℎ ∈ (𝑥 RingHom 𝑦)))
11		rhmisrnghm 20372	. . . . . 6 ⊢ (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥 RngHom 𝑦))
12	7	sseld 3981	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ 𝑆))
13	7	sseld 3981	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑅 → 𝑦 ∈ 𝑆))
14	12, 13	anim12d 608	. . . . . . . . 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 7577	. . . . . . . 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 2818	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦) ↔ ℎ ∈ (𝑥 RngHom 𝑦)))
19	11, 18	imbitrrid 245	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦)))
20	10, 19	sylbid 239	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦)))
21	20	ssrdv 3988	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
22	21	ralrimivva 3199	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
23		inss1 4228	. . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring
24	5, 23	eqsstrdi 4036	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring)
25		xpss12 5691	. . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
26	24, 24, 25	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27		rhmfn 20391	. . . . 5 ⊢ RingHom Fn (Ring × Ring)
28		fnssresb 6672	. . . . 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 4228	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
32	6, 31	eqsstrdi 4036	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ Rng)
33		xpss12 5691	. . . . 5 ⊢ ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
34	32, 32, 33	syl2anc 583	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35		rnghmfn 20331	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
36		fnssresb 6672	. . . . 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 4201	. . . . . 6 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41		inex1g 5319	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
42	40, 41	eqeltrid 2836	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (Rng ∩ 𝑈) ∈ V)
43	39, 42	syl 17	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V)
44	6, 43	eqeltrd 2832	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
45	30, 38, 44	isssc 17772	. 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 710	1 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ (𝑆 × 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  Vcvv 3473   ∩ cin 3947   ⊆ wss 3948   class class class wbr 5148   × cxp 5674   ↾ cres 5678   Fn wfn 6538  (class class class)co 7412   ⊆_cat cssc 17759  Rngcrng 20047  Ringcrg 20128   RngHom crnghm 20326   RingHom crh 20361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-plusg 17215  df-0g 17392  df-ssc 17762  df-mgm 18566  df-mgmhm 18618  df-sgrp 18645  df-mnd 18661  df-mhm 18706  df-grp 18859  df-minusg 18860  df-ghm 19129  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-rnghm 20328  df-rhm 20364

This theorem is referenced by:  rhmsubcrngc  47016  rhmsubc  47077  rhmsubcALTV  47095