Theorem rhmsscrnghm 20717

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 20337	. . . . . 6 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3	2	ssrdv 3944	. . . 4 ⊢ (𝜑 → Ring ⊆ Rng)
4	3	ssrind 4197	. . 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 3993	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
8		ovres 7564	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
9	8	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
10	9	eleq2d 2850	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ℎ ∈ (𝑥 RingHom 𝑦)))
11		rhmisrnghm 20531	. . . . . 6 ⊢ (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥 RngHom 𝑦))
12	7	sseld 3937	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ 𝑆))
13	7	sseld 3937	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ 𝑅 → 𝑦 ∈ 𝑆))
14	12, 13	anim12d 618	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)))
15	14	imp 410	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆))
16		ovres 7564	. . . . . . . 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 2850	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦) ↔ ℎ ∈ (𝑥 RngHom 𝑦)))
19	11, 18	imbitrrid 248	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦)))
20	10, 19	sylbid 242	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ℎ ∈ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦)))
21	20	ssrdv 3944	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
22	21	ralrimivva 3207	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
23		inss1 4190	. . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring
24	5, 23	eqsstrdi 3982	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring)
25		xpss12 5664	. . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
26	24, 24, 25	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27		rhmfn 20550	. . . . 5 ⊢ RingHom Fn (Ring × Ring)
28		fnssresb 6645	. . . . 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 259	. . 3 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
31		inss1 4190	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
32	6, 31	eqsstrdi 3982	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ Rng)
33		xpss12 5664	. . . . 5 ⊢ ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
34	32, 32, 33	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35		rnghmfn 20490	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
36		fnssresb 6645	. . . . 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 259	. . 3 ⊢ (𝜑 → ( RngHom ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
39		rhmsscrnghm.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
40		incom 4163	. . . . . 6 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41		inex1g 5277	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
42	40, 41	eqeltrid 2868	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (Rng ∩ 𝑈) ∈ V)
43	39, 42	syl 17	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V)
44	6, 43	eqeltrd 2864	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
45	30, 38, 44	isssc 17855	. 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 723	1 ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ (𝑆 × 𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∀wral 3078  Vcvv 3456   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5102   × cxp 5647   ↾ cres 5651   Fn wfn 6518  (class class class)co 7398   ⊆_cat cssc 17842  Rngcrng 20200  Ringcrg 20285   RngHom crnghm 20485   RingHom crh 20520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720  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 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-er 8680  df-map 8812  df-ixp 8882  df-en 8930  df-dom 8931  df-sdom 8932  df-pnf 11220  df-mnf 11221  df-xr 11222  df-ltxr 11223  df-le 11224  df-sub 11418  df-neg 11419  df-nn 12213  df-2 12282  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17248  df-plusg 17301  df-0g 17472  df-ssc 17845  df-mgm 18676  df-mgmhm 18728  df-sgrp 18755  df-mnd 18771  df-mhm 18819  df-grp 18980  df-minusg 18981  df-ghm 19256  df-cmn 19824  df-abl 19825  df-mgp 20189  df-rng 20201  df-ur 20234  df-ring 20287  df-rnghm 20487  df-rhm 20523

This theorem is referenced by:  rhmsubcrngc  20720  rhmsubc  20741  rhmsubcALTV  48912