lkrpssN - Mathbox for Norm Megill

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Theorem lkrpssN 39965

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 3924	. . 3 ⊢ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)))
2		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐺) ⊊ (𝐾‘𝐻))
3		eqid 2762	. . . . . . . . . 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 21238	. . . . . . . . . . 11 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
8	6, 7	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ LMod)
9		lkrpss.h	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ∈ 𝐹)
10	3, 4, 5, 8, 9	lkrssv 39898	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝐻) ⊆ (Base‘𝑊))
11	10	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
12	2, 11	psssstrd 4066	. . . . . . 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 606	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
15		simplr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
16		eqid 2762	. . . . . . . . . . 11 ⊢ (LSHyp‘𝑊) = (LSHyp‘𝑊)
17	6	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → 𝑊 ∈ LVec)
18		simpr 489	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ∈ (LSHyp‘𝑊))
19		simplr 780	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
20	10	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐻) ⊆ (Base‘𝑊))
21		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) = (Base‘𝑊))
22		simpllr 787	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
23	21, 22	eqsstrrd 3971	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → (Base‘𝑊) ⊆ (𝐾‘𝐻))
24	20, 23	eqssd 3953	. . . . . . . . . . . . . 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 39910	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝐾‘𝐻) ≠ (Base‘𝑊) ↔ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
26	25	ad3antrrr 742	. . . . . . . . . . . . . . 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 2996	. . . . . . . . . . . . . 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 260	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) ∧ (𝐾‘𝐺) = (Base‘𝑊)) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
29	19, 28	pm2.21dd 198	. . . . . . . . . . . 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 39909	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
32	31	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ∈ (LSHyp‘𝑊) ∨ (𝐾‘𝐺) = (Base‘𝑊)))
33	18, 29, 32	mpjaodan 972	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) ∈ (LSHyp‘𝑊))
34		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
35	16, 17, 33, 34	lshpcmp 39790	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) = (𝐾‘𝐻)))
36	15, 35	mpbid 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) ∧ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)) → (𝐾‘𝐺) = (𝐾‘𝐻))
37	36	ex 417	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ((𝐾‘𝐻) ∈ (LSHyp‘𝑊) → (𝐾‘𝐺) = (𝐾‘𝐻)))
38	37	necon3ad 2970	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ((𝐾‘𝐺) ≠ (𝐾‘𝐻) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊)))
39	38	impr 459	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → ¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊))
40	25	necon1bbid 2996	. . . . . . 7 ⊢ (𝜑 → (¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊) ↔ (𝐾‘𝐻) = (Base‘𝑊)))
41	40	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (¬ (𝐾‘𝐻) ∈ (LSHyp‘𝑊) ↔ (𝐾‘𝐻) = (Base‘𝑊)))
42	39, 41	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻))) → (𝐾‘𝐻) = (Base‘𝑊))
43	14, 42	jca 520	. . . 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 39898	. . . . . . 7 ⊢ (𝜑 → (𝐾‘𝐺) ⊆ (Base‘𝑊))
45	44	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (Base‘𝑊))
46		simprr 784	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐻) = (Base‘𝑊))
47	46	eqcomd 2768	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (Base‘𝑊) = (𝐾‘𝐻))
48	45, 47	sseqtrd 3972	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))
49		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (Base‘𝑊))
50	49, 47	neeqtrd 3026	. . . . 5 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → (𝐾‘𝐺) ≠ (𝐾‘𝐻))
51	48, 50	jca 520	. . . 4 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)))
52	43, 51	impbida 812	. . 3 ⊢ (𝜑 → (((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ∧ (𝐾‘𝐺) ≠ (𝐾‘𝐻)) ↔ ((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊))))
53	1, 52	bitrid 286	. 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 39963	. . . 4 ⊢ (𝜑 → ((𝐾‘𝐺) = (Base‘𝑊) ↔ 𝐺 = 0 ))
57	56	necon3bid 3001	. . 3 ⊢ (𝜑 → ((𝐾‘𝐺) ≠ (Base‘𝑊) ↔ 𝐺 ≠ 0 ))
58	3, 4, 5, 54, 55, 8, 9	lkr0f2 39963	. . 3 ⊢ (𝜑 → ((𝐾‘𝐻) = (Base‘𝑊) ↔ 𝐻 = 0 ))
59	57, 58	anbi12d 643	. 2 ⊢ (𝜑 → (((𝐾‘𝐺) ≠ (Base‘𝑊) ∧ (𝐾‘𝐻) = (Base‘𝑊)) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))
60	53, 59	bitrd 282	1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ 0 ∧ 𝐻 = 0 )))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ⊆ wss 3904 ⊊ wpss 3905 ‘cfv 6536 Basecbs 17275 0_gc0g 17498 LModclmod 20992 LVecclvec 21234 LSHypclsh 39777 LFnlclfn 39859 LKerclk 39887 LDualcld 39925

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-tpos 8220 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-er 8692 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-n0 12511 df-z 12598 df-uz 12869 df-fz 13542 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-sca 17332 df-vsca 17333 df-0g 17500 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-grp 19009 df-minusg 19010 df-sbg 19011 df-subg 19195 df-cntz 19393 df-lsm 19712 df-cmn 19858 df-abl 19859 df-mgp 20223 df-rng 20237 df-ur 20270 df-ring 20323 df-oppr 20426 df-dvdsr 20446 df-unit 20447 df-invr 20477 df-drng 20840 df-lmod 20994 df-lss 21064 df-lsp 21104 df-lvec 21235 df-lshyp 39779 df-lfl 39860 df-lkr 39888 df-ldual 39926

This theorem is used by: lkrss2N 39971 lkreqN 39972

W3C validator