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Theorem rhmsscrnghm 46924
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 46655 . . . . . 6 (𝑟 ∈ Ring → 𝑟 ∈ Rng)
21a1i 11 . . . . 5 (𝜑 → (𝑟 ∈ Ring → 𝑟 ∈ Rng))
32ssrdv 3989 . . . 4 (𝜑 → Ring ⊆ Rng)
43ssrind 4236 . . 3 (𝜑 → (Ring ∩ 𝑈) ⊆ (Rng ∩ 𝑈))
5 rhmsscrnghm.r . . 3 (𝜑𝑅 = (Ring ∩ 𝑈))
6 rhmsscrnghm.s . . 3 (𝜑𝑆 = (Rng ∩ 𝑈))
74, 5, 63sstr4d 4030 . 2 (𝜑𝑅𝑆)
8 ovres 7573 . . . . . . 7 ((𝑥𝑅𝑦𝑅) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
98adantl 483 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) = (𝑥 RingHom 𝑦))
109eleq2d 2820 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ↔ ∈ (𝑥 RingHom 𝑦)))
11 rhmisrnghm 46723 . . . . . 6 ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥 RngHomo 𝑦))
127sseld 3982 . . . . . . . . . 10 (𝜑 → (𝑥𝑅𝑥𝑆))
137sseld 3982 . . . . . . . . . 10 (𝜑 → (𝑦𝑅𝑦𝑆))
1412, 13anim12d 610 . . . . . . . . 9 (𝜑 → ((𝑥𝑅𝑦𝑅) → (𝑥𝑆𝑦𝑆)))
1514imp 408 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥𝑆𝑦𝑆))
16 ovres 7573 . . . . . . . 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 2820 . . . . . 6 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦) ↔ ∈ (𝑥 RngHomo 𝑦)))
1911, 18imbitrrid 245 . . . . 5 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥 RingHom 𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2010, 19sylbid 239 . . . 4 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → ( ∈ (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) → ∈ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦)))
2120ssrdv 3989 . . 3 ((𝜑 ∧ (𝑥𝑅𝑦𝑅)) → (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
2221ralrimivva 3201 . 2 (𝜑 → ∀𝑥𝑅𝑦𝑅 (𝑥( RingHom ↾ (𝑅 × 𝑅))𝑦) ⊆ (𝑥( RngHomo ↾ (𝑆 × 𝑆))𝑦))
23 inss1 4229 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
245, 23eqsstrdi 4037 . . . . 5 (𝜑𝑅 ⊆ Ring)
25 xpss12 5692 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
2624, 24, 25syl2anc 585 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
27 rhmfn 46721 . . . . 5 RingHom Fn (Ring × Ring)
28 fnssresb 6673 . . . . 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 257 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
31 inss1 4229 . . . . . 6 (Rng ∩ 𝑈) ⊆ Rng
326, 31eqsstrdi 4037 . . . . 5 (𝜑𝑆 ⊆ Rng)
33 xpss12 5692 . . . . 5 ((𝑆 ⊆ Rng ∧ 𝑆 ⊆ Rng) → (𝑆 × 𝑆) ⊆ (Rng × Rng))
3432, 32, 33syl2anc 585 . . . 4 (𝜑 → (𝑆 × 𝑆) ⊆ (Rng × Rng))
35 rnghmfn 46688 . . . . 5 RngHomo Fn (Rng × Rng)
36 fnssresb 6673 . . . . 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 257 . . 3 (𝜑 → ( RngHomo ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
39 rhmsscrnghm.u . . . . 5 (𝜑𝑈𝑉)
40 incom 4202 . . . . . 6 (Rng ∩ 𝑈) = (𝑈 ∩ Rng)
41 inex1g 5320 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Rng) ∈ V)
4240, 41eqeltrid 2838 . . . . 5 (𝑈𝑉 → (Rng ∩ 𝑈) ∈ V)
4339, 42syl 17 . . . 4 (𝜑 → (Rng ∩ 𝑈) ∈ V)
446, 43eqeltrd 2834 . . 3 (𝜑𝑆 ∈ V)
4530, 38, 44isssc 17767 . 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 205  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cin 3948  wss 3949   class class class wbr 5149   × cxp 5675  cres 5679   Fn wfn 6539  (class class class)co 7409  cat cssc 17754  Ringcrg 20056   RingHom crh 20248  Rngcrng 46648   RngHomo crngh 46683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-ssc 17757  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-grp 18822  df-minusg 18823  df-ghm 19090  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-rnghom 20251  df-mgmhm 46549  df-rng 46649  df-rnghomo 46685
This theorem is referenced by:  rhmsubcrngc  46927  rhmsubc  46988  rhmsubcALTV  47006
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