lkrpssN - Mathbox for Norm Megill

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Theorem lkrpssN 37104

Description: Proper subset relation between kernels. (Contributed by NM, 16-Feb-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
lkrpss.f	⊢ 𝐹 = (LFnl‘𝑊)
lkrpss.k	⊢ 𝐾 = (LKer‘𝑊)
lkrpss.d	⊢ 𝐷 = (LDual‘𝑊)
lkrpss.o	⊢ 0 = (0_g‘𝐷)
lkrpss.w	⊢ (𝜑 → 𝑊 ∈ LVec)
lkrpss.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lkrpss.h	⊢ (𝜑 → 𝐻 ∈ 𝐹)

**Assertion**
Ref	Expression
lkrpssN	⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))

**Proof of Theorem lkrpssN**
Step	Hyp	Ref	Expression
1		df-pss 3902	. . 3 ⊢ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)))
2		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ⊊ (𝐾‘𝐻))
3		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑊) = (Base‘𝑊)
4		lkrpss.f	. . . . . . . . . 10 ⊢ 𝐹 = (LFnl‘𝑊)
5		lkrpss.k	. . . . . . . . . 10 ⊢ 𝐾 = (LKer‘𝑊)
6		lkrpss.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LVec)
7		lveclmod 20283	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
8	6, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod)
9		lkrpss.h	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
10	3, 4, 5, 8, 9	lkrssv 37037	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝐻) ⊆ (Base‘𝑊))
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
12	2, 11	psssstrd 4040	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ⊊ (Base‘𝑊))
13	12	pssned 4029	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ≠ (Base‘𝑊))
14	1, 13	sylan2br 594	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
15		simplr 765	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
16		eqid 2738	. . . . . . . . . . 11 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
17	6	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → 𝑊 ∈ LVec)
18		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ∈ (LSHyp‘𝑊))
19		simplr 765	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
20	10	ad3antrrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
21		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) = (Base‘𝑊))
22		simpllr 772	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
23	21, 22	eqsstrrd 3956	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (Base‘𝑊) ⊆ (𝐾‘𝐻))
24	20, 23	eqssd 3934	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) = (Base‘𝑊))
25	3, 16, 4, 5, 6, 9	lkrshp4 37049	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐾‘𝐻) ≠ (Base‘𝑊) ↔ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
26	25	ad3antrrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → ((𝐾‘𝐻) ≠ (Base‘𝑊) ↔ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
27	26	necon1bbid 2982	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊) ↔ (𝐾‘𝐻) = (Base‘𝑊)))
28	24, 27	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
29	19, 28	pm2.21dd 194	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) ∈ (LSHyp‘𝑊))
30		lkrpss.g	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
31	3, 16, 4, 5, 6, 30	lkrshpor 37048	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
32	31	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
33	18, 29, 32	mpjaodan 955	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ∈ (LSHyp‘𝑊))
34		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
35	16, 17, 33, 34	lshpcmp 36929	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) = (𝐾‘𝐻)))
36	15, 35	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) = (𝐾‘𝐻))
37	36	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ((𝐾‘𝐻) ∈ (LSHyp‘𝑊) → (𝐾‘𝐺) = (𝐾‘𝐻)))
38	37	necon3ad 2955	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ((𝐾‘𝐺) ≠ (𝐾‘𝐻) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
39	38	impr 454	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
40	25	necon1bbid 2982	. . . . . . 7 ⊢ (𝜑 → (¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊) ↔ (𝐾‘𝐻) = (Base‘𝑊)))
41	40	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊) ↔ (𝐾‘𝐻) = (Base‘𝑊)))
42	39, 41	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐻) = (Base‘𝑊))
43	14, 42	jca 511	. . . 4 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊)))
44	3, 4, 5, 8, 30	lkrssv 37037	. . . . . . 7 ⊢ (𝜑 → (𝐾‘𝐺) ⊆ (Base‘𝑊))
45	44	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (Base‘𝑊))
46		simprr 769	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐻) = (Base‘𝑊))
47	46	eqcomd 2744	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (Base‘𝑊) = (𝐾‘𝐻))
48	45, 47	sseqtrd 3957	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
49		simprl 767	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
50	49, 47	neeqtrd 3012	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (𝐾‘𝐻))
51	48, 50	jca 511	. . . 4 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)))
52	43, 51	impbida 797	. . 3 ⊢ (𝜑 → (((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)) ↔ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))))
53	1, 52	syl5bb 282	. 2 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))))
54		lkrpss.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑊)
55		lkrpss.o	. . . . 5 ⊢ 0 = (0_g‘𝐷)
56	3, 4, 5, 54, 55, 8, 30	lkr0f2 37102	. . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) = (Base‘𝑊) ↔ 𝐺 = 0 ))
57	56	necon3bid 2987	. . 3 ⊢ (𝜑 → ((𝐾‘𝐺) ≠ (Base‘𝑊) ↔ 𝐺 ≠ 0 ))
58	3, 4, 5, 54, 55, 8, 9	lkr0f2 37102	. . 3 ⊢ (𝜑 → ((𝐾‘𝐻) = (Base‘𝑊) ↔ 𝐻 = 0 ))
59	57, 58	anbi12d 630	. 2 ⊢ (𝜑 → (((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊)) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))
60	53, 59	bitrd 278	1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ⊆ wss 3883 ⊊ wpss 3884 ‘cfv 6418 Basecbs 16840 0_gc0g 17067 LModclmod 20038 LVecclvec 20279 LSHypclsh 36916 LFnlclfn 36998 LKerclk 37026 LDualcld 37064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lshyp 36918 df-lfl 36999 df-lkr 37027 df-ldual 37065

This theorem is referenced by: lkrss2N 37110 lkreqN 37111

W3C validator