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Theorem rhmsscrnghm 41817
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 41668 . . . . . 6 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
21a1i 11 . . . . 5 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
32ssrdv 3570 . . . 4 (𝜑 → Ring ⊆ Rng)
4 ssrin 3796 . . . 4 (Ring ⊆ Rng → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
53, 4syl 17 . . 3 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
6 rhmsscrnghm.r . . 3 (𝜑𝑅 = (Ring ∩ 𝑈))
7 rhmsscrnghm.s . . 3 (𝜑𝑆 = (Rng ∩ 𝑈))
85, 6, 73sstr4d 3607 . 2 (𝜑𝑅𝑆)
9 ovres 6673 . . . . . . 7 ((𝑥𝑅𝑦𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
109adantl 480 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
1110eleq2d 2669 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ∈ (𝑥 RingHom 𝑦)))
12 rhmisrnghm 41709 . . . . . 6 ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥 RngHomo 𝑦))
138sseld 3563 . . . . . . . . . 10 (𝜑 → (𝑥𝑅𝑥𝑆))
148sseld 3563 . . . . . . . . . 10 (𝜑 → (𝑦𝑅𝑦𝑆))
1513, 14anim12d 583 . . . . . . . . 9 (𝜑 → ((𝑥𝑅𝑦𝑅) → (𝑥𝑆𝑦𝑆)))
1615imp 443 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥𝑆𝑦𝑆))
17 ovres 6673 . . . . . . . 8 ((𝑥𝑆𝑦𝑆) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
1816, 17syl 17 . . . . . . 7 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) = (𝑥 RngHomo 𝑦))
1918eleq2d 2669 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) ↔ ∈ (𝑥 RngHomo 𝑦)))
2012, 19syl5ibr 234 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2111, 20sylbid 228 . . . 4 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2221ssrdv 3570 . . 3 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
2322ralrimivva 2950 . 2 (𝜑 → ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
24 inss1 3791 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
256, 24syl6eqss 3614 . . . . 5 (𝜑𝑅 ⊆ Ring)
26 xpss12 5134 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
2725, 25, 26syl2anc 690 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
28 rhmfn 41707 . . . . 5 RingHom Fn (Ring × Ring)
29 fnssresb 5900 . . . . 5 ( RingHom Fn (Ring × Ring) → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
3028, 29mp1i 13 . . . 4 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅) ↔ (𝑅 × 𝑅) ⊆ (Ring × Ring)))
3127, 30mpbird 245 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
32 inss1 3791 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
337, 32syl6eqss 3614 . . . . 5 (𝜑𝑆 ⊆ Rng)
34 xpss12 5134 . . . . 5 ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
3533, 33, 34syl2anc 690 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
36 rnghmfn 41679 . . . . 5 RngHomo Fn (Rng × Rng)
37 fnssresb 5900 . . . . 5 ( RngHomo Fn (Rng × Rng) → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
3836, 37mp1i 13 . . . 4 (𝜑 → (( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆) ↔ (𝑆 × 𝑆) ⊆ (Rng × Rng)))
3935, 38mpbird 245 . . 3 (𝜑 → ( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
40 rhmsscrnghm.u . . . . 5 (𝜑𝑈𝑉)
41 incom 3763 . . . . . 6 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
42 inex1g 4721 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4341, 42syl5eqel 2688 . . . . 5 (𝑈𝑉 → (Rng ∩ 𝑈) ∈ V)
4440, 43syl 17 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
457, 44eqeltrd 2684 . . 3 (𝜑𝑆 ∈ V)
4631, 39, 45isssc 16246 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)) ↔ (𝑅𝑆 ∧ ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))))
478, 23, 46mpbir2and 958 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2892  Vcvv 3169  cin 3535  wss 3536   class class class wbr 4574   × cxp 5023  cres 5027   Fn wfn 5782  (class class class)co 6524  cat cssc 16233  Ringcrg 18313   RingHom crh 18478  Rngcrng 41663   RngHomo crngh 41674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-map 7720  df-ixp 7769  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-ndx 15641  df-slot 15642  df-base 15643  df-sets 15644  df-plusg 15724  df-0g 15868  df-ssc 16236  df-mgm 17008  df-sgrp 17050  df-mnd 17061  df-mhm 17101  df-grp 17191  df-minusg 17192  df-ghm 17424  df-cmn 17961  df-abl 17962  df-mgp 18256  df-ur 18268  df-ring 18315  df-rnghom 18481  df-mgmhm 41568  df-rng0 41664  df-rnghomo 41676
This theorem is referenced by:  rhmsubcrngc  41820  rhmsubc  41881  rhmsubcALTV  41900
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