Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmsscrnghm Structured version   Visualization version   GIF version

Theorem rhmsscrnghm 44581
 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 44434 . . . . . 6 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
21a1i 11 . . . . 5 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
32ssrdv 3959 . . . 4 (𝜑 → Ring ⊆ Rng)
43ssrind 4197 . . 3 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
5 rhmsscrnghm.r . . 3 (𝜑𝑅 = (Ring ∩ 𝑈))
6 rhmsscrnghm.s . . 3 (𝜑𝑆 = (Rng ∩ 𝑈))
74, 5, 63sstr4d 4000 . 2 (𝜑𝑅𝑆)
8 ovres 7308 . . . . . . 7 ((𝑥𝑅𝑦𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
98adantl 485 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
109eleq2d 2901 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ∈ (𝑥 RingHom 𝑦)))
11 rhmisrnghm 44475 . . . . . 6 ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥 RngHomo 𝑦))
127sseld 3952 . . . . . . . . . 10 (𝜑 → (𝑥𝑅𝑥𝑆))
137sseld 3952 . . . . . . . . . 10 (𝜑 → (𝑦𝑅𝑦𝑆))
1412, 13anim12d 611 . . . . . . . . 9 (𝜑 → ((𝑥𝑅𝑦𝑅) → (𝑥𝑆𝑦𝑆)))
1514imp 410 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥𝑆𝑦𝑆))
16 ovres 7308 . . . . . . . 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 2901 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) ↔ ∈ (𝑥 RngHomo 𝑦)))
1911, 18syl5ibr 249 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2010, 19sylbid 243 . . . 4 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2120ssrdv 3959 . . 3 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
2221ralrimivva 3186 . 2 (𝜑 → ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
23 inss1 4190 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
245, 23eqsstrdi 4007 . . . . 5 (𝜑𝑅 ⊆ Ring)
25 xpss12 5557 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
2624, 24, 25syl2anc 587 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27 rhmfn 44473 . . . . 5 RingHom Fn (Ring × Ring)
28 fnssresb 6458 . . . . 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 260 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
31 inss1 4190 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
326, 31eqsstrdi 4007 . . . . 5 (𝜑𝑆 ⊆ Rng)
33 xpss12 5557 . . . . 5 ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
3432, 32, 33syl2anc 587 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35 rnghmfn 44445 . . . . 5 RngHomo Fn (Rng × Rng)
36 fnssresb 6458 . . . . 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 260 . . 3 (𝜑 → ( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
39 rhmsscrnghm.u . . . . 5 (𝜑𝑈𝑉)
40 incom 4163 . . . . . 6 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41 inex1g 5209 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4240, 41eqeltrid 2920 . . . . 5 (𝑈𝑉 → (Rng ∩ 𝑈) ∈ V)
4339, 42syl 17 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
446, 43eqeltrd 2916 . . 3 (𝜑𝑆 ∈ V)
4530, 38, 44isssc 17090 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)) ↔ (𝑅𝑆 ∧ ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))))
467, 22, 45mpbir2and 712 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ∩ cin 3918   ⊆ wss 3919   class class class wbr 5052   × cxp 5540   ↾ cres 5544   Fn wfn 6338  (class class class)co 7149   ⊆cat cssc 17077  Ringcrg 19297   RingHom crh 19467  Rngcrng 44429   RngHomo crngh 44440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-map 8404  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-plusg 16578  df-0g 16715  df-ssc 17080  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-mhm 17956  df-grp 18106  df-minusg 18107  df-ghm 18356  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-rnghom 19470  df-mgmhm 44330  df-rng0 44430  df-rnghomo 44442 This theorem is referenced by:  rhmsubcrngc  44584  rhmsubc  44645  rhmsubcALTV  44663
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