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Theorem resrhm2b 20548
Description: Restriction of the codomain of a (ring) homomorphism. resghm2b 19195 analog. (Contributed by SN, 7-Feb-2025.)
Hypothesis
Ref Expression
resrhm2b.u π‘ˆ = (𝑇 β†Ύs 𝑋)
Assertion
Ref Expression
resrhm2b ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom π‘ˆ)))
Proof of Theorem resrhm2b
StepHypRef Expression
1 subrgsubg 20523 . . . . . 6 (𝑋 ∈ (SubRingβ€˜π‘‡) β†’ 𝑋 ∈ (SubGrpβ€˜π‘‡))
2 resrhm2b.u . . . . . . 7 π‘ˆ = (𝑇 β†Ύs 𝑋)
32resghm2b 19195 . . . . . 6 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom π‘ˆ)))
41, 3sylan 578 . . . . 5 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom π‘ˆ)))
5 eqid 2728 . . . . . . . 8 (mulGrpβ€˜π‘‡) = (mulGrpβ€˜π‘‡)
65subrgsubm 20531 . . . . . . 7 (𝑋 ∈ (SubRingβ€˜π‘‡) β†’ 𝑋 ∈ (SubMndβ€˜(mulGrpβ€˜π‘‡)))
7 eqid 2728 . . . . . . . 8 ((mulGrpβ€˜π‘‡) β†Ύs 𝑋) = ((mulGrpβ€˜π‘‡) β†Ύs 𝑋)
87resmhm2b 18781 . . . . . . 7 ((𝑋 ∈ (SubMndβ€˜(mulGrpβ€˜π‘‡)) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡)) ↔ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom ((mulGrpβ€˜π‘‡) β†Ύs 𝑋))))
96, 8sylan 578 . . . . . 6 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡)) ↔ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom ((mulGrpβ€˜π‘‡) β†Ύs 𝑋))))
10 subrgrcl 20522 . . . . . . . . . 10 (𝑋 ∈ (SubRingβ€˜π‘‡) β†’ 𝑇 ∈ Ring)
112, 5mgpress 20096 . . . . . . . . . 10 ((𝑇 ∈ Ring ∧ 𝑋 ∈ (SubRingβ€˜π‘‡)) β†’ ((mulGrpβ€˜π‘‡) β†Ύs 𝑋) = (mulGrpβ€˜π‘ˆ))
1210, 11mpancom 686 . . . . . . . . 9 (𝑋 ∈ (SubRingβ€˜π‘‡) β†’ ((mulGrpβ€˜π‘‡) β†Ύs 𝑋) = (mulGrpβ€˜π‘ˆ))
1312adantr 479 . . . . . . . 8 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ ((mulGrpβ€˜π‘‡) β†Ύs 𝑋) = (mulGrpβ€˜π‘ˆ))
1413oveq2d 7442 . . . . . . 7 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ ((mulGrpβ€˜π‘†) MndHom ((mulGrpβ€˜π‘‡) β†Ύs 𝑋)) = ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ)))
1514eleq2d 2815 . . . . . 6 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom ((mulGrpβ€˜π‘‡) β†Ύs 𝑋)) ↔ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ))))
169, 15bitrd 278 . . . . 5 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡)) ↔ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ))))
174, 16anbi12d 630 . . . 4 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡))) ↔ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ)))))
1817anbi2d 628 . . 3 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡)))) ↔ (𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ))))))
1910adantr 479 . . . . 5 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ 𝑇 ∈ Ring)
2019biantrud 530 . . . 4 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ 𝑇 ∈ Ring)))
2120anbi1d 629 . . 3 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡)))) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡))))))
222subrgring 20520 . . . . . 6 (𝑋 ∈ (SubRingβ€˜π‘‡) β†’ π‘ˆ ∈ Ring)
2322adantr 479 . . . . 5 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ π‘ˆ ∈ Ring)
2423biantrud 530 . . . 4 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝑆 ∈ Ring ↔ (𝑆 ∈ Ring ∧ π‘ˆ ∈ Ring)))
2524anbi1d 629 . . 3 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ ((𝑆 ∈ Ring ∧ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ)))) ↔ ((𝑆 ∈ Ring ∧ π‘ˆ ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ))))))
2618, 21, 253bitr3d 308 . 2 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡)))) ↔ ((𝑆 ∈ Ring ∧ π‘ˆ ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ))))))
27 eqid 2728 . . 3 (mulGrpβ€˜π‘†) = (mulGrpβ€˜π‘†)
2827, 5isrhm 20424 . 2 (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ ((𝑆 ∈ Ring ∧ 𝑇 ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘‡)))))
29 eqid 2728 . . 3 (mulGrpβ€˜π‘ˆ) = (mulGrpβ€˜π‘ˆ)
3027, 29isrhm 20424 . 2 (𝐹 ∈ (𝑆 RingHom π‘ˆ) ↔ ((𝑆 ∈ Ring ∧ π‘ˆ ∈ Ring) ∧ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ∧ 𝐹 ∈ ((mulGrpβ€˜π‘†) MndHom (mulGrpβ€˜π‘ˆ)))))
3126, 28, 303bitr4g 313 1 ((𝑋 ∈ (SubRingβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   βŠ† wss 3949  ran crn 5683  β€˜cfv 6553  (class class class)co 7426   β†Ύs cress 17216   MndHom cmhm 18745  SubMndcsubmnd 18746  SubGrpcsubg 19082   GrpHom cghm 19174  mulGrpcmgp 20081  Ringcrg 20180   RingHom crh 20415  SubRingcsubrg 20513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mhm 18747  df-submnd 18748  df-grp 18900  df-minusg 18901  df-subg 19085  df-ghm 19175  df-mgp 20082  df-ur 20129  df-ring 20182  df-rhm 20418  df-subrg 20515
This theorem is referenced by:  imadrhmcl  20692  idomsubr  33020  algextdeglem4  33421  selvcllem4  41845
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