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Theorem rhmsscrnghm 42873
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 (𝜑𝑈𝑉)
rhmsscrnghm.r (𝜑𝑅 = (Ring ∩ 𝑈))
rhmsscrnghm.s (𝜑𝑆 = (Rng ∩ 𝑈))
Assertion
Ref Expression
rhmsscrnghm (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))

Proof of Theorem rhmsscrnghm
Dummy variables 𝑥 𝑦 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringrng 42726 . . . . . 6 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
21a1i 11 . . . . 5 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
32ssrdv 3833 . . . 4 (𝜑 → Ring ⊆ Rng)
43ssrind 4064 . . 3 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
5 rhmsscrnghm.r . . 3 (𝜑𝑅 = (Ring ∩ 𝑈))
6 rhmsscrnghm.s . . 3 (𝜑𝑆 = (Rng ∩ 𝑈))
74, 5, 63sstr4d 3873 . 2 (𝜑𝑅𝑆)
8 ovres 7060 . . . . . . 7 ((𝑥𝑅𝑦𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
98adantl 475 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
109eleq2d 2892 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ∈ (𝑥 RingHom 𝑦)))
11 rhmisrnghm 42767 . . . . . 6 ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥 RngHomo 𝑦))
127sseld 3826 . . . . . . . . . 10 (𝜑 → (𝑥𝑅𝑥𝑆))
137sseld 3826 . . . . . . . . . 10 (𝜑 → (𝑦𝑅𝑦𝑆))
1412, 13anim12d 604 . . . . . . . . 9 (𝜑 → ((𝑥𝑅𝑦𝑅) → (𝑥𝑆𝑦𝑆)))
1514imp 397 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥𝑆𝑦𝑆))
16 ovres 7060 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
1715, 16syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
1817eleq2d 2892 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) ↔ ∈ (𝑥 RngHomo 𝑦)))
1911, 18syl5ibr 238 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2010, 19sylbid 232 . . . 4 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2120ssrdv 3833 . . 3 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
2221ralrimivva 3180 . 2 (𝜑 → ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
23 inss1 4057 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
245, 23syl6eqss 3880 . . . . 5 (𝜑𝑅 ⊆ Ring)
25 xpss12 5357 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
2624, 24, 25syl2anc 581 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27 rhmfn 42765 . . . . 5 RingHom Fn (Ring × Ring)
28 fnssresb 6236 . . . . 5 ( RingHom Fn (Ring × Ring) → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
2927, 28mp1i 13 . . . 4 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
3026, 29mpbird 249 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
31 inss1 4057 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
326, 31syl6eqss 3880 . . . . 5 (𝜑𝑆 ⊆ Rng)
33 xpss12 5357 . . . . 5 ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
3432, 32, 33syl2anc 581 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35 rnghmfn 42737 . . . . 5 RngHomo Fn (Rng × Rng)
36 fnssresb 6236 . . . . 5 ( RngHomo Fn (Rng × Rng) → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
3735, 36mp1i 13 . . . 4 (𝜑 → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
3834, 37mpbird 249 . . 3 (𝜑 → ( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
39 rhmsscrnghm.u . . . . 5 (𝜑𝑈𝑉)
40 incom 4032 . . . . . 6 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41 inex1g 5026 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4240, 41syl5eqel 2910 . . . . 5 (𝑈𝑉 → (Rng ∩ 𝑈) ∈ V)
4339, 42syl 17 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
446, 43eqeltrd 2906 . . 3 (𝜑𝑆 ∈ V)
4530, 38, 44isssc 16832 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)) ↔ (𝑅𝑆 ∧ ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))))
467, 22, 45mpbir2and 706 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3117  Vcvv 3414  cin 3797  wss 3798   class class class wbr 4873   × cxp 5340  cres 5344   Fn wfn 6118  (class class class)co 6905  cat cssc 16819  Ringcrg 18901   RingHom crh 19068  Rngcrng 42721   RngHomo crngh 42732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-map 8124  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-plusg 16318  df-0g 16455  df-ssc 16822  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-grp 17779  df-minusg 17780  df-ghm 18009  df-cmn 18548  df-abl 18549  df-mgp 18844  df-ur 18856  df-ring 18903  df-rnghom 19071  df-mgmhm 42626  df-rng0 42722  df-rnghomo 42734
This theorem is referenced by:  rhmsubcrngc  42876  rhmsubc  42937  rhmsubcALTV  42955
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