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Theorem resmhm2b 17721
 Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resmhm2.u 𝑈 = (𝑇s 𝑋)
Assertion
Ref Expression
resmhm2b ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))

Proof of Theorem resmhm2b
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmrcl1 17698 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
21adantl 475 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
3 resmhm2.u . . . . . 6 𝑈 = (𝑇s 𝑋)
43submmnd 17714 . . . . 5 (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 ∈ Mnd)
54ad2antrr 717 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑈 ∈ Mnd)
62, 5jca 507 . . 3 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd))
7 eqid 2825 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
8 eqid 2825 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
97, 8mhmf 17700 . . . . . . . 8 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
109adantl 475 . . . . . . 7 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1110ffnd 6283 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
12 simplr 785 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ran 𝐹𝑋)
13 df-f 6131 . . . . . 6 (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹𝑋))
1411, 12, 13sylanbrc 578 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
153submbas 17715 . . . . . . 7 (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
1615ad2antrr 717 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑋 = (Base‘𝑈))
1716feq3d 6269 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
1814, 17mpbid 224 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
19 eqid 2825 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
20 eqid 2825 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
217, 19, 20mhmlin 17702 . . . . . . . 8 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
22213expb 1153 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2322adantll 705 . . . . . 6 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
243, 20ressplusg 16359 . . . . . . . 8 (𝑋 ∈ (SubMnd‘𝑇) → (+g𝑇) = (+g𝑈))
2524ad3antrrr 721 . . . . . . 7 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g𝑇) = (+g𝑈))
2625oveqd 6927 . . . . . 6 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2723, 26eqtrd 2861 . . . . 5 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2827ralrimivva 3180 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
29 eqid 2825 . . . . . . 7 (0g𝑆) = (0g𝑆)
30 eqid 2825 . . . . . . 7 (0g𝑇) = (0g𝑇)
3129, 30mhm0 17703 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
3231adantl 475 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
333, 30subm0 17716 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑇) → (0g𝑇) = (0g𝑈))
3433ad2antrr 717 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (0g𝑇) = (0g𝑈))
3532, 34eqtrd 2861 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑈))
3618, 28, 353jca 1162 . . 3 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑈)))
37 eqid 2825 . . . 4 (Base‘𝑈) = (Base‘𝑈)
38 eqid 2825 . . . 4 (+g𝑈) = (+g𝑈)
39 eqid 2825 . . . 4 (0g𝑈) = (0g𝑈)
407, 37, 19, 38, 29, 39ismhm 17697 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑈) ↔ ((𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd) ∧ (𝐹:(Base‘𝑆)⟶(Base‘𝑈) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑈))))
416, 36, 40sylanbrc 578 . 2 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑈))
423resmhm2 17720 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4342ancoms 452 . . 3 ((𝑋 ∈ (SubMnd‘𝑇) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4443adantlr 706 . 2 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4541, 44impbida 835 1 ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117   ⊆ wss 3798  ran crn 5347   Fn wfn 6122  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  Basecbs 16229   ↾s cress 16230  +gcplusg 16312  0gc0g 16460  Mndcmnd 17654   MndHom cmhm 17693  SubMndcsubmnd 17694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-ndx 16232  df-slot 16233  df-base 16235  df-sets 16236  df-ress 16237  df-plusg 16325  df-0g 16462  df-mgm 17602  df-sgrp 17644  df-mnd 17655  df-mhm 17695  df-submnd 17696 This theorem is referenced by:  resghm2b  18036  m2cpmmhm  20927  dchrghm  25401  lgseisenlem4  25523
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