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Theorem resmhm2b 17408
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 17385 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝑆 ∈ Mnd)
21adantl 481 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑆 ∈ Mnd)
3 resmhm2.u . . . . . 6 𝑈 = (𝑇s 𝑋)
43submmnd 17401 . . . . 5 (𝑋 ∈ (SubMnd‘𝑇) → 𝑈 ∈ Mnd)
54ad2antrr 762 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑈 ∈ Mnd)
62, 5jca 553 . . 3 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝑆 ∈ Mnd ∧ 𝑈 ∈ Mnd))
7 eqid 2651 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
8 eqid 2651 . . . . . . . . 9 (Base‘𝑇) = (Base‘𝑇)
97, 8mhmf 17387 . . . . . . . 8 (𝐹 ∈ (𝑆 MndHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
109adantl 481 . . . . . . 7 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
11 ffn 6083 . . . . . . 7 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
1210, 11syl 17 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 Fn (Base‘𝑆))
13 simplr 807 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ran 𝐹𝑋)
14 df-f 5930 . . . . . 6 (𝐹:(Base‘𝑆)⟶𝑋 ↔ (𝐹 Fn (Base‘𝑆) ∧ ran 𝐹𝑋))
1512, 13, 14sylanbrc 699 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶𝑋)
163submbas 17402 . . . . . . 7 (𝑋 ∈ (SubMnd‘𝑇) → 𝑋 = (Base‘𝑈))
1716ad2antrr 762 . . . . . 6 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝑋 = (Base‘𝑈))
1817feq3d 6070 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹:(Base‘𝑆)⟶𝑋𝐹:(Base‘𝑆)⟶(Base‘𝑈)))
1915, 18mpbid 222 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑈))
20 eqid 2651 . . . . . . . . 9 (+g𝑆) = (+g𝑆)
21 eqid 2651 . . . . . . . . 9 (+g𝑇) = (+g𝑇)
227, 20, 21mhmlin 17389 . . . . . . . 8 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
23223expb 1285 . . . . . . 7 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
2423adantll 750 . . . . . 6 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑇)(𝐹𝑦)))
253, 21ressplusg 16040 . . . . . . . 8 (𝑋 ∈ (SubMnd‘𝑇) → (+g𝑇) = (+g𝑈))
2625ad3antrrr 766 . . . . . . 7 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (+g𝑇) = (+g𝑈))
2726oveqd 6707 . . . . . 6 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → ((𝐹𝑥)(+g𝑇)(𝐹𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2824, 27eqtrd 2685 . . . . 5 ((((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆))) → (𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
2928ralrimivva 3000 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → ∀𝑥 ∈ (Base‘𝑆)∀𝑦 ∈ (Base‘𝑆)(𝐹‘(𝑥(+g𝑆)𝑦)) = ((𝐹𝑥)(+g𝑈)(𝐹𝑦)))
30 eqid 2651 . . . . . . 7 (0g𝑆) = (0g𝑆)
31 eqid 2651 . . . . . . 7 (0g𝑇) = (0g𝑇)
3230, 31mhm0 17390 . . . . . 6 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹‘(0g𝑆)) = (0g𝑇))
3332adantl 481 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑇))
343, 31subm0 17403 . . . . . 6 (𝑋 ∈ (SubMnd‘𝑇) → (0g𝑇) = (0g𝑈))
3534ad2antrr 762 . . . . 5 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (0g𝑇) = (0g𝑈))
3633, 35eqtrd 2685 . . . 4 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → (𝐹‘(0g𝑆)) = (0g𝑈))
3719, 29, 363jca 1261 . . 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)))
38 eqid 2651 . . . 4 (Base‘𝑈) = (Base‘𝑈)
39 eqid 2651 . . . 4 (+g𝑈) = (+g𝑈)
40 eqid 2651 . . . 4 (0g𝑈) = (0g𝑈)
417, 38, 20, 39, 30, 40ismhm 17384 . . 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))))
426, 37, 41sylanbrc 699 . 2 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑈))
433resmhm2 17407 . . . 4 ((𝐹 ∈ (𝑆 MndHom 𝑈) ∧ 𝑋 ∈ (SubMnd‘𝑇)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4443ancoms 468 . . 3 ((𝑋 ∈ (SubMnd‘𝑇) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4544adantlr 751 . 2 (((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) ∧ 𝐹 ∈ (𝑆 MndHom 𝑈)) → 𝐹 ∈ (𝑆 MndHom 𝑇))
4642, 45impbida 895 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wss 3607  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  s cress 15905  +gcplusg 15988  0gc0g 16147  Mndcmnd 17341   MndHom cmhm 17380  SubMndcsubmnd 17381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383
This theorem is referenced by:  resghm2b  17725  m2cpmmhm  20598  dchrghm  25026  lgseisenlem4  25148
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