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Theorem resrhm2b 20493
Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 19149 analog. (Contributed by SN, 7-Feb-2025.)
Hypothesis
Ref Expression
resrhm2b.u 𝑈 = (𝑇s 𝑋)
Assertion
Ref Expression
resrhm2b ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))

Proof of Theorem resrhm2b
StepHypRef Expression
1 subrgsubg 20468 . . . . . 6 (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubGrp‘𝑇))
2 resrhm2b.u . . . . . . 7 𝑈 = (𝑇s 𝑋)
32resghm2b 19149 . . . . . 6 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
41, 3sylan 579 . . . . 5 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
5 eqid 2731 . . . . . . . 8 (mulGrp‘𝑇) = (mulGrp‘𝑇)
65subrgsubm 20476 . . . . . . 7 (𝑋 ∈ (SubRing‘𝑇) → 𝑋 ∈ (SubMnd‘(mulGrp‘𝑇)))
7 eqid 2731 . . . . . . . 8 ((mulGrp‘𝑇) ↾s 𝑋) = ((mulGrp‘𝑇) ↾s 𝑋)
87resmhm2b 18740 . . . . . . 7 ((𝑋 ∈ (SubMnd‘(mulGrp‘𝑇)) ∧ ran 𝐹𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋))))
96, 8sylan 579 . . . . . 6 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋))))
10 subrgrcl 20467 . . . . . . . . . 10 (𝑋 ∈ (SubRing‘𝑇) → 𝑇 ∈ Ring)
112, 5mgpress 20044 . . . . . . . . . 10 ((𝑇 ∈ Ring ∧ 𝑋 ∈ (SubRing‘𝑇)) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
1210, 11mpancom 685 . . . . . . . . 9 (𝑋 ∈ (SubRing‘𝑇) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
1312adantr 480 . . . . . . . 8 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → ((mulGrp‘𝑇) ↾s 𝑋) = (mulGrp‘𝑈))
1413oveq2d 7428 . . . . . . 7 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋)) = ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))
1514eleq2d 2818 . . . . . 6 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom ((mulGrp‘𝑇) ↾s 𝑋)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))
169, 15bitrd 279 . . . . 5 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)) ↔ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))
174, 16anbi12d 630 . . . 4 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇))) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))))
1817anbi2d 628 . . 3 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))) ↔ (𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))))
1910adantr 480 . . . . 5 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → 𝑇 ∈ Ring)
2019biantrud 531 . . . 4 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ 𝑇 ∈ Ring)))
2120anbi1d 629 . . 3 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇))))))
222subrgring 20465 . . . . . 6 (𝑋 ∈ (SubRing‘𝑇) → 𝑈 ∈ Ring)
2322adantr 480 . . . . 5 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → 𝑈 ∈ Ring)
2423biantrud 531 . . . 4 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ 𝑈 ∈ Ring)))
2524anbi1d 629 . . 3 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))) ↔ ((𝑆 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))))
2618, 21, 253bitr3d 309 . 2 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))) ↔ ((𝑆 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈))))))
27 eqid 2731 . . 3 (mulGrp‘𝑆) = (mulGrp‘𝑆)
2827, 5isrhm 20370 . 2 (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑇)))))
29 eqid 2731 . . 3 (mulGrp‘𝑈) = (mulGrp‘𝑈)
3027, 29isrhm 20370 . 2 (𝐹 ∈ (𝑆 RingHom 𝑈) ↔ ((𝑆 ∈ Ring ∧ 𝑈 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑈) ∧ 𝐹 ∈ ((mulGrp‘𝑆) MndHom (mulGrp‘𝑈)))))
3126, 28, 303bitr4g 314 1 ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wss 3948  ran crn 5677  cfv 6543  (class class class)co 7412  s cress 17178   MndHom cmhm 18704  SubMndcsubmnd 18705  SubGrpcsubg 19037   GrpHom cghm 19128  mulGrpcmgp 20029  Ringcrg 20128   RingHom crh 20361  SubRingcsubrg 20458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18706  df-submnd 18707  df-grp 18859  df-minusg 18860  df-subg 19040  df-ghm 19129  df-mgp 20030  df-ur 20077  df-ring 20130  df-rhm 20364  df-subrg 20460
This theorem is referenced by:  imadrhmcl  20557  algextdeglem4  33066  selvcllem4  41456
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