Theorem rhmsscrnghm 20682

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 20299	. . . . . 6 ⊢ (𝑟 ∈ Ring → 𝑟 ∈ Rng)
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
3	2	ssrdv 4001	. . . 4 ⊢ (𝜑 → Ring ⊆ Rng)
4	3	ssrind 4252	. . 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 4043	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑆)
8		ovres 7599	. . . . . . 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 2825	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (ℎ ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ℎ ∈ (𝑥 RingHom 𝑦)))
11		rhmisrnghm 20497	. . . . . 6 ⊢ (ℎ ∈ (𝑥 RingHom 𝑦) → ℎ ∈ (𝑥 RngHom 𝑦))
12	7	sseld 3994	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝑅 → 𝑥 ∈ 𝑆))
13	7	sseld 3994	. . . . . . . . . 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 7599	. . . . . . . 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 2825	. . . . . 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 4001	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
22	21	ralrimivva 3200	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHom ↾ (𝑆 × 𝑆))𝑦))
23		inss1 4245	. . . . . 6 ⊢ (Ring ∩ 𝑈) ⊆ Ring
24	5, 23	eqsstrdi 4050	. . . . 5 ⊢ (𝜑 → 𝑅 ⊆ Ring)
25		xpss12 5704	. . . . 5 ⊢ ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
26	24, 24, 25	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27		rhmfn 20516	. . . . 5 ⊢ RingHom Fn (Ring × Ring)
28		fnssresb 6691	. . . . 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 4245	. . . . . 6 ⊢ (Rng ∩ 𝑈) ⊆ Rng
32	6, 31	eqsstrdi 4050	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ Rng)
33		xpss12 5704	. . . . 5 ⊢ ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
34	32, 32, 33	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35		rnghmfn 20456	. . . . 5 ⊢ RngHom Fn (Rng × Rng)
36		fnssresb 6691	. . . . 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 4217	. . . . . 6 ⊢ (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41		inex1g 5325	. . . . . 6 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∩ Rng) ∈ V)
42	40, 41	eqeltrid 2843	. . . . 5 ⊢ (𝑈 ∈ 𝑉 → (Rng ∩ 𝑈) ∈ V)
43	39, 42	syl 17	. . . 4 ⊢ (𝜑 → (Rng ∩ 𝑈) ∈ V)
44	6, 43	eqeltrd 2839	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
45	30, 38, 44	isssc 17868	. 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 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ∩ cin 3962   ⊆ wss 3963   class class class wbr 5148   × cxp 5687   ↾ cres 5691   Fn wfn 6558  (class class class)co 7431   ⊆_cat cssc 17855  Rngcrng 20170  Ringcrg 20251   RngHom crnghm 20451   RingHom crh 20486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-0g 17488  df-ssc 17858  df-mgm 18666  df-mgmhm 18718  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-rnghm 20453  df-rhm 20489

This theorem is referenced by:  rhmsubcrngc  20685  rhmsubc  20706  rhmsubcALTV  48129