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Theorem lmhmpreima 19407
 Description: The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
lmhmima.x 𝑋 = (LSubSp‘𝑆)
lmhmima.y 𝑌 = (LSubSp‘𝑇)
Assertion
Ref Expression
lmhmpreima ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)

Proof of Theorem lmhmpreima
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lmghm 19390 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
21adantr 474 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3 lmhmlmod2 19391 . . . 4 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
4 lmhmima.y . . . . 5 𝑌 = (LSubSp‘𝑇)
54lsssubg 19316 . . . 4 ((𝑇 ∈ LMod ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
63, 5sylan 577 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
7 ghmpreima 18033 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
82, 6, 7syl2anc 581 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ (SubGrp‘𝑆))
9 lmhmlmod1 19392 . . . . . 6 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
109ad2antrr 719 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑆 ∈ LMod)
11 simprl 789 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
12 cnvimass 5726 . . . . . . . 8 (𝐹𝑈) ⊆ dom 𝐹
13 eqid 2825 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑆)
14 eqid 2825 . . . . . . . . . 10 (Base‘𝑇) = (Base‘𝑇)
1513, 14lmhmf 19393 . . . . . . . . 9 (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1615adantr 474 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
1712, 16fssdm 6294 . . . . . . 7 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ⊆ (Base‘𝑆))
1817sselda 3827 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑏 ∈ (𝐹𝑈)) → 𝑏 ∈ (Base‘𝑆))
1918adantrl 709 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (Base‘𝑆))
20 eqid 2825 . . . . . 6 (Scalar‘𝑆) = (Scalar‘𝑆)
21 eqid 2825 . . . . . 6 ( ·𝑠𝑆) = ( ·𝑠𝑆)
22 eqid 2825 . . . . . 6 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2313, 20, 21, 22lmodvscl 19236 . . . . 5 ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
2410, 11, 19, 23syl3anc 1496 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆))
25 simpll 785 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
26 eqid 2825 . . . . . . 7 ( ·𝑠𝑇) = ( ·𝑠𝑇)
2720, 22, 13, 21, 26lmhmlin 19394 . . . . . 6 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
2825, 11, 19, 27syl3anc 1496 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) = (𝑎( ·𝑠𝑇)(𝐹𝑏)))
293ad2antrr 719 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑇 ∈ LMod)
30 simplr 787 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑈𝑌)
31 eqid 2825 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
3220, 31lmhmsca 19389 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
3332adantr 474 . . . . . . . . . 10 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
3433fveq2d 6437 . . . . . . . . 9 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
3534eleq2d 2892 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
3635biimpar 471 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3736adantrr 710 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
3816ffund 6282 . . . . . . . 8 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → Fun 𝐹)
3938adantr 474 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → Fun 𝐹)
40 simprr 791 . . . . . . 7 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → 𝑏 ∈ (𝐹𝑈))
41 fvimacnvi 6580 . . . . . . 7 ((Fun 𝐹𝑏 ∈ (𝐹𝑈)) → (𝐹𝑏) ∈ 𝑈)
4239, 40, 41syl2anc 581 . . . . . 6 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹𝑏) ∈ 𝑈)
43 eqid 2825 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
4431, 26, 43, 4lssvscl 19314 . . . . . 6 (((𝑇 ∈ LMod ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹𝑏) ∈ 𝑈)) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4529, 30, 37, 42, 44syl22anc 874 . . . . 5 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑇)(𝐹𝑏)) ∈ 𝑈)
4628, 45eqeltrd 2906 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)
47 ffn 6278 . . . . . 6 (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
48 elpreima 6586 . . . . . 6 (𝐹 Fn (Base‘𝑆) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
4916, 47, 483syl 18 . . . . 5 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
5049adantr 474 . . . 4 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → ((𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈) ↔ ((𝑎( ·𝑠𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠𝑆)𝑏)) ∈ 𝑈)))
5124, 46, 50mpbir2and 706 . . 3 (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (𝐹𝑈))) → (𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
5251ralrimivva 3180 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))
539adantr 474 . . 3 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → 𝑆 ∈ LMod)
54 lmhmima.x . . . 4 𝑋 = (LSubSp‘𝑆)
5520, 22, 13, 21, 54islss4 19321 . . 3 (𝑆 ∈ LMod → ((𝐹𝑈) ∈ 𝑋 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))))
5653, 55syl 17 . 2 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → ((𝐹𝑈) ∈ 𝑋 ↔ ((𝐹𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (𝐹𝑈)(𝑎( ·𝑠𝑆)𝑏) ∈ (𝐹𝑈))))
578, 52, 56mpbir2and 706 1 ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∀wral 3117  ◡ccnv 5341   “ cima 5345  Fun wfun 6117   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  Scalarcsca 16308   ·𝑠 cvsca 16309  SubGrpcsubg 17939   GrpHom cghm 18008  LModclmod 19219  LSubSpclss 19288   LMHom clmhm 19378 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  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 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-grp 17779  df-minusg 17780  df-sbg 17781  df-subg 17942  df-ghm 18009  df-mgp 18844  df-ur 18856  df-ring 18903  df-lmod 19221  df-lss 19289  df-lmhm 19381 This theorem is referenced by:  lmhmlsp  19408  lmhmkerlss  19410  lnmepi  38498
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