Theorem lmhmpreima 21043

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 21026	. . 3 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		lmhmlmod2 21027	. . . 4 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
3		lmhmima.y	. . . . 5 ⊢ 𝑌 = (LSubSp‘𝑇)
4	3	lsssubg 20952	. . . 4 ⊢ ((𝑇 ∈ LMod ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
5	2, 4	sylan 581	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑈 ∈ (SubGrp‘𝑇))
6		ghmpreima 19213	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑈 ∈ (SubGrp‘𝑇)) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
7	1, 5, 6	syl2an2r 686	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ (SubGrp‘𝑆))
8		lmhmlmod1 21028	. . . . . 6 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
9	8	ad2antrr 727	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑆 ∈ LMod)
10		simprl 771	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑆)))
11		cnvimass 6047	. . . . . . . 8 ⊢ (◡𝐹 “ 𝑈) ⊆ dom 𝐹
12		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑇) = (Base‘𝑇)
14	12, 13	lmhmf 21029	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
15	14	adantr 480	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
16	11, 15	fssdm 6687	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ⊆ (Base‘𝑆))
17	16	sselda 3921	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → 𝑏 ∈ (Base‘𝑆))
18	17	adantrl 717	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (Base‘𝑆))
19		eqid 2736	. . . . . 6 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆)
20		eqid 2736	. . . . . 6 ⊢ ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘𝑆)
21		eqid 2736	. . . . . 6 ⊢ (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
22	12, 19, 20, 21	lmodvscl 20873	. . . . 5 ⊢ ((𝑆 ∈ LMod ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (Base‘𝑆)) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
23	9, 10, 18, 22	syl3anc 1374	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (Base‘𝑆))
24		simpll 767	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝐹 ∈ (𝑆 LMHom 𝑇))
25		eqid 2736	. . . . . . 7 ⊢ ( ·𝑠 ‘𝑇) = ( ·𝑠 ‘𝑇)
26	19, 21, 12, 20, 25	lmhmlin 21030	. . . . . 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 1374	. . . . 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 727	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑇 ∈ LMod)
29		simplr 769	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑈 ∈ 𝑌)
30		eqid 2736	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇)
31	19, 30	lmhmsca 21025	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
32	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Scalar‘𝑇) = (Scalar‘𝑆))
33	32	fveq2d 6844	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
34	33	eleq2d 2822	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (𝑎 ∈ (Base‘(Scalar‘𝑇)) ↔ 𝑎 ∈ (Base‘(Scalar‘𝑆))))
35	34	biimpar 477	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ 𝑎 ∈ (Base‘(Scalar‘𝑆))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
36	35	adantrr 718	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑎 ∈ (Base‘(Scalar‘𝑇)))
37	15	ffund 6672	. . . . . . 7 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → Fun 𝐹)
38		simprr 773	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → 𝑏 ∈ (◡𝐹 “ 𝑈))
39		fvimacnvi 7004	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑏 ∈ (◡𝐹 “ 𝑈)) → (𝐹‘𝑏) ∈ 𝑈)
40	37, 38, 39	syl2an2r 686	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘𝑏) ∈ 𝑈)
41		eqid 2736	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
42	30, 25, 41, 3	lssvscl 20950	. . . . . 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 839	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑇)(𝐹‘𝑏)) ∈ 𝑈)
44	27, 43	eqeltrd 2836	. . . 4 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝐹‘(𝑎( ·𝑠 ‘𝑆)𝑏)) ∈ 𝑈)
45		ffn 6668	. . . . . 6 ⊢ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) → 𝐹 Fn (Base‘𝑆))
46		elpreima 7010	. . . . . 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 18	. . . . 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 480	. . . 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 714	. . 3 ⊢ (((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) ∧ (𝑎 ∈ (Base‘(Scalar‘𝑆)) ∧ 𝑏 ∈ (◡𝐹 “ 𝑈))) → (𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
50	49	ralrimivva 3180	. 2 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → ∀𝑎 ∈ (Base‘(Scalar‘𝑆))∀𝑏 ∈ (◡𝐹 “ 𝑈)(𝑎( ·𝑠 ‘𝑆)𝑏) ∈ (◡𝐹 “ 𝑈))
51	8	adantr 480	. . 3 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → 𝑆 ∈ LMod)
52		lmhmima.x	. . . 4 ⊢ 𝑋 = (LSubSp‘𝑆)
53	19, 21, 12, 20, 52	islss4 20957	. . 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 17	. 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 714	1 ⊢ ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈 ∈ 𝑌) → (◡𝐹 “ 𝑈) ∈ 𝑋)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ^◡ccnv 5630   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Scalarcsca 17223   ·_𝑠 cvsca 17224  SubGrpcsubg 19096   GrpHom cghm 19187  LModclmod 20855  LSubSpclss 20926   LMHom clmhm 21014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-ghm 19188  df-mgp 20122  df-ur 20163  df-ring 20216  df-lmod 20857  df-lss 20927  df-lmhm 21017

This theorem is referenced by:  lmhmlsp  21044  lmhmkerlss  21046  lnmepi  43513