lkrpssN - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrpssN		Structured version Visualization version GIF version

Theorem lkrpssN 39149

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 3931	. . 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 2729	. . . . . . . . . 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 21045	. . . . . . . . . . 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 39082	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝐻) ⊆ (Base‘𝑊))
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
12	2, 11	psssstrd 4071	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ⊊ (Base‘𝑊))
13	12	pssned 4060	. . . . . 6 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ≠ (Base‘𝑊))
14	1, 13	sylan2br 595	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
15		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
16		eqid 2729	. . . . . . . . . . 11 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
17	6	ad2antrr 726	. . . . . . . . . . 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 768	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
20	10	ad3antrrr 730	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
23	21, 22	eqsstrrd 3979	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (Base‘𝑊) ⊆ (𝐾‘𝐻))
24	20, 23	eqssd 3961	. . . . . . . . . . . . . 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 39094	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐾‘𝐻) ≠ (Base‘𝑊) ↔ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
26	25	ad3antrrr 730	. . . . . . . . . . . . . . 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 2964	. . . . . . . . . . . . . 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 39093	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
32	31	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
33	18, 29, 32	mpjaodan 960	. . . . . . . . . . 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 38974	. . . . . . . . . 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 2938	. . . . . . 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 2964	. . . . . . 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 39082	. . . . . . 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 772	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐻) = (Base‘𝑊))
47	46	eqcomd 2735	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (Base‘𝑊) = (𝐾‘𝐻))
48	45, 47	sseqtrd 3980	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
49		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
50	49, 47	neeqtrd 2994	. . . . 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 800	. . 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 39147	. . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) = (Base‘𝑊) ↔ 𝐺 = 0 ))
57	56	necon3bid 2969	. . 3 ⊢ (𝜑 → ((𝐾‘𝐺) ≠ (Base‘𝑊) ↔ 𝐺 ≠ 0 ))
58	3, 4, 5, 54, 55, 8, 9	lkr0f2 39147	. . 3 ⊢ (𝜑 → ((𝐾‘𝐻) = (Base‘𝑊) ↔ 𝐻 = 0 ))
59	57, 58	anbi12d 632	. 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 847 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ⊆ wss 3911 ⊊ wpss 3912 ‘cfv 6499 Basecbs 17155 0_gc0g 17378 LModclmod 20798 LVecclvec 21041 LSHypclsh 38961 LFnlclfn 39043 LKerclk 39071 LDualcld 39109

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-n0 12419 df-z 12506 df-uz 12770 df-fz 13445 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-0g 17380 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-subg 19037 df-cntz 19231 df-lsm 19550 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-drng 20651 df-lmod 20800 df-lss 20870 df-lsp 20910 df-lvec 21042 df-lshyp 38963 df-lfl 39044 df-lkr 39072 df-ldual 39110

This theorem is referenced by: lkrss2N 39155 lkreqN 39156

W3C validator