lkrpssN - Mathbox for Norm Megill

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Theorem lkrpssN 39460

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 3922	. . 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 2737	. . . . . . . . . 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 21062	. . . . . . . . . . 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 39393	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝐻) ⊆ (Base‘𝑊))
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
12	2, 11	psssstrd 4065	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ⊊ (Base‘𝑊))
13	12	pssned 4054	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ≠ (Base‘𝑊))
14	1, 13	sylan2br 596	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
15		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
16		eqid 2737	. . . . . . . . . . 11 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
17	6	ad2antrr 727	. . . . . . . . . . 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 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
20	10	ad3antrrr 731	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
23	21, 22	eqsstrrd 3970	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (Base‘𝑊) ⊆ (𝐾‘𝐻))
24	20, 23	eqssd 3952	. . . . . . . . . . . . . 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 39405	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐾‘𝐻) ≠ (Base‘𝑊) ↔ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
26	25	ad3antrrr 731	. . . . . . . . . . . . . . 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 2972	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
29	19, 28	pm2.21dd 195	. . . . . . . . . . . 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 39404	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
32	31	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
33	18, 29, 32	mpjaodan 961	. . . . . . . . . . 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 39285	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) = (𝐾‘𝐻)))
36	15, 35	mpbid 232	. . . . . . . . 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 2946	. . . . . . 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 2972	. . . . . . 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 232	. . . . 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 39393	. . . . . . 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 773	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐻) = (Base‘𝑊))
47	46	eqcomd 2743	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (Base‘𝑊) = (𝐾‘𝐻))
48	45, 47	sseqtrd 3971	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
49		simprl 771	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
50	49, 47	neeqtrd 3002	. . . . 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 801	. . 3 ⊢ (𝜑 → (((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)) ↔ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))))
53	1, 52	bitrid 283	. 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 39458	. . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) = (Base‘𝑊) ↔ 𝐺 = 0 ))
57	56	necon3bid 2977	. . 3 ⊢ (𝜑 → ((𝐾‘𝐺) ≠ (Base‘𝑊) ↔ 𝐺 ≠ 0 ))
58	3, 4, 5, 54, 55, 8, 9	lkr0f2 39458	. . 3 ⊢ (𝜑 → ((𝐾‘𝐻) = (Base‘𝑊) ↔ 𝐻 = 0 ))
59	57, 58	anbi12d 633	. 2 ⊢ (𝜑 → (((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊)) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))
60	53, 59	bitrd 279	1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ⊆ wss 3902 ⊊ wpss 3903 ‘cfv 6493 Basecbs 17140 0_gc0g 17363 LModclmod 20815 LVecclvec 21058 LSHypclsh 39272 LFnlclfn 39354 LKerclk 39382 LDualcld 39420

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 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-0g 17365 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-subg 19057 df-cntz 19250 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-drng 20668 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lvec 21059 df-lshyp 39274 df-lfl 39355 df-lkr 39383 df-ldual 39421

This theorem is referenced by: lkrss2N 39466 lkreqN 39467

W3C validator