lkrpssN - Mathbox for Norm Megill

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Theorem lkrpssN 39792

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 3925	. . 3 ⊢ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)))
2		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ⊊ (𝐾‘𝐻))
3		eqid 2763	. . . . . . . . . 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 21180	. . . . . . . . . . 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 39725	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝐻) ⊆ (Base‘𝑊))
11	10	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
12	2, 11	psssstrd 4067	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ⊊ (Base‘𝑊))
13	12	pssned 4055	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ≠ (Base‘𝑊))
14	1, 13	sylan2br 604	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
15		simplr 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
16		eqid 2763	. . . . . . . . . . 11 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
17	6	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → 𝑊 ∈ LVec)
18		simpr 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ∈ (LSHyp‘𝑊))
19		simplr 778	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
20	10	ad3antrrr 740	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
21		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) = (Base‘𝑊))
22		simpllr 785	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
23	21, 22	eqsstrrd 3972	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (Base‘𝑊) ⊆ (𝐾‘𝐻))
24	20, 23	eqssd 3954	. . . . . . . . . . . . . 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 39737	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐾‘𝐻) ≠ (Base‘𝑊) ↔ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
26	25	ad3antrrr 740	. . . . . . . . . . . . . . 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 2997	. . . . . . . . . . . . . 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 259	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
29	19, 28	pm2.21dd 197	. . . . . . . . . . . 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 39736	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
32	31	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
33	18, 29, 32	mpjaodan 971	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ∈ (LSHyp‘𝑊))
34		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
35	16, 17, 33, 34	lshpcmp 39617	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) = (𝐾‘𝐻)))
36	15, 35	mpbid 234	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) = (𝐾‘𝐻))
37	36	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ((𝐾‘𝐻) ∈ (LSHyp‘𝑊) → (𝐾‘𝐺) = (𝐾‘𝐻)))
38	37	necon3ad 2971	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ((𝐾‘𝐺) ≠ (𝐾‘𝐻) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
39	38	impr 458	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
40	25	necon1bbid 2997	. . . . . . 7 ⊢ (𝜑 → (¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊) ↔ (𝐾‘𝐻) = (Base‘𝑊)))
41	40	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊) ↔ (𝐾‘𝐻) = (Base‘𝑊)))
42	39, 41	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐻) = (Base‘𝑊))
43	14, 42	jca 519	. . . 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 39725	. . . . . . 7 ⊢ (𝜑 → (𝐾‘𝐺) ⊆ (Base‘𝑊))
45	44	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (Base‘𝑊))
46		simprr 782	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐻) = (Base‘𝑊))
47	46	eqcomd 2769	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (Base‘𝑊) = (𝐾‘𝐻))
48	45, 47	sseqtrd 3973	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
49		simprl 780	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
50	49, 47	neeqtrd 3027	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (𝐾‘𝐻))
51	48, 50	jca 519	. . . 4 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)))
52	43, 51	impbida 810	. . 3 ⊢ (𝜑 → (((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)) ↔ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))))
53	1, 52	bitrid 285	. 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 39790	. . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) = (Base‘𝑊) ↔ 𝐺 = 0 ))
57	56	necon3bid 3002	. . 3 ⊢ (𝜑 → ((𝐾‘𝐺) ≠ (Base‘𝑊) ↔ 𝐺 ≠ 0 ))
58	3, 4, 5, 54, 55, 8, 9	lkr0f2 39790	. . 3 ⊢ (𝜑 → ((𝐾‘𝐻) = (Base‘𝑊) ↔ 𝐻 = 0 ))
59	57, 58	anbi12d 641	. 2 ⊢ (𝜑 → (((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊)) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))
60	53, 59	bitrd 281	1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ⊆ wss 3905 ⊊ wpss 3906 ‘cfv 6521 Basecbs 17255 0_gc0g 17478 LModclmod 20934 LVecclvec 21176 LSHypclsh 39604 LFnlclfn 39686 LKerclk 39714 LDualcld 39752

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-2 12290 df-3 12291 df-4 12292 df-5 12293 df-6 12294 df-n0 12492 df-z 12579 df-uz 12850 df-fz 13523 df-struct 17193 df-sets 17210 df-slot 17228 df-ndx 17240 df-base 17256 df-ress 17277 df-plusg 17309 df-mulr 17310 df-sca 17312 df-vsca 17313 df-0g 17480 df-mgm 18684 df-sgrp 18763 df-mnd 18779 df-submnd 18828 df-grp 18988 df-minusg 18989 df-sbg 18990 df-subg 19175 df-cntz 19367 df-lsm 19686 df-cmn 19832 df-abl 19833 df-mgp 20197 df-rng 20209 df-ur 20242 df-ring 20295 df-oppr 20396 df-dvdsr 20416 df-unit 20417 df-invr 20447 df-drng 20790 df-lmod 20936 df-lss 21006 df-lsp 21046 df-lvec 21177 df-lshyp 39606 df-lfl 39687 df-lkr 39715 df-ldual 39753

This theorem is referenced by: lkrss2N 39798 lkreqN 39799

W3C validator