Theorem lmhmpreima 21374

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) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ 𝑋)

Proof of Theorem lmhmpreima

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lmghm 21357	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod2 21358	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3		lmhmima.y	. . . . 5 ⊢ 𝑌 = (LSubSp‘𝑇)
4	3	lsssubg 21283	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
5	2, 4	sylan 592	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
6		ghmpreima 19502	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
7	1, 5, 6	syl2an2r 698	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
8		lmhmlmod1 21359	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
9	8	ad2antrr 739	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑆 ∈ LMod)
10		simprl 783	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
11		cnvimass 6217	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
12		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
14	12, 13	lmhmf 21360	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
15	14	adantr 486	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16	11, 15	fssdm 6759	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ⊆ (Base‘𝑆))
17	16	sselda 3913	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → 𝑏 ∈ (Base‘𝑆))
18	17	adantrl 729	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (Base‘𝑆))
19		eqid 2763	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
20		eqid 2763	. . . . . 6 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
21		eqid 2763	. . . . . 6 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
22	12, 19, 20, 21	lmodvscl 21204	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
23	9, 10, 18, 22	syl3anc 1399	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
24		simpll 779	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
25		eqid 2763	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
26	19, 21, 12, 20, 25	lmhmlin 21361	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
27	24, 10, 18, 26	syl3anc 1399	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) = (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)))
28	2	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑇 ∈ LMod)
29		simplr 781	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑈 ∈ 𝑌)
30		eqid 2763	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
31	19, 30	lmhmsca 21356	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
32	31	adantr 486	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
33	32	fveq2d 6920	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
34	33	eleq2d 2850	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
35	34	biimpar 483	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
36	35	adantrr 730	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
37	15	ffund 6744	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → Fun 𝐹)
38		simprr 785	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (◡𝐹 “ 𝑈))
39		fvimacnvi 7082	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → (𝐹‘𝑏) ∈ 𝑈)
40	37, 38, 39	syl2an2r 698	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘𝑏) ∈ 𝑈)
41		eqid 2763	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
42	30, 25, 41, 3	lssvscl 21281	. . . . . 6 ⊢ (((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑇)) ∧ (𝐹‘𝑏) ∈ 𝑈)) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)) ∈ 𝑈)
43	28, 29, 36, 40, 42	syl22anc 852	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)) ∈ 𝑈)
44	27, 43	eqeltrd 2864	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)
45		ffn 6739	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
46		elpreima 7088	. . . . . 6 ⊢ (𝐹 Fn (Base‘𝑆) → ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈) ↔ ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)))
47	15, 45, 46	3syl 19	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈) ↔ ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)))
48	47	adantr 486	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈) ↔ ((𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆) ∧ (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)))
49	23, 44, 48	mpbir2and 726	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
50	49	ralrimivva 3209	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
51	8	adantr 486	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑆 ∈ LMod)
52		lmhmima.x	. . . 4 ⊢ 𝑋 = (LSubSp‘𝑆)
53	19, 21, 12, 20, 52	islss4 21288	. . 3 ⊢ (𝑆 ∈ LMod → ((◡𝐹 “ 𝑈) ∈ 𝑋 ↔ ((◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))))
54	51, 53	syl 18	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ((◡𝐹 “ 𝑈) ∈ 𝑋 ↔ ((◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆) ∧ ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))))
55	7, 50, 54	mpbir2and 726	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ 𝑋)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ^◡ccnv 5657   “ cima 5661  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  Scalarcsca 17475   ·_𝑠 cvsca 17476  SubGrpcsubg 19380   GrpHom cghm 19477  LModclmod 21186  LSubSpclss 21257   LMHom clmhm 21345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-sets 17386  df-slot 17404  df-ndx 17416  df-base 17432  df-ress 17453  df-plusg 17486  df-0g 17662  df-mgm 18866  df-sgrp 18958  df-mnd 18974  df-grp 19197  df-minusg 19198  df-sbg 19199  df-subg 19383  df-ghm 19478  df-mgp 20412  df-ur 20459  df-ring 20512  df-lmod 21188  df-lss 21258  df-lmhm 21348

This theorem is used by:  lmhmlsp  21375  lmhmkerlss  21377  lnmepi  44353