Metamath Proof Explorer


Theorem lkrpssN

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

Ref Expression
Hypotheses lkrpss.f 𝐹 = ( LFnl ‘ 𝑊 )
lkrpss.k 𝐾 = ( LKer ‘ 𝑊 )
lkrpss.d 𝐷 = ( LDual ‘ 𝑊 )
lkrpss.o 0 = ( 0g𝐷 )
lkrpss.w ( 𝜑𝑊 ∈ LVec )
lkrpss.g ( 𝜑𝐺𝐹 )
lkrpss.h ( 𝜑𝐻𝐹 )
Assertion lkrpssN ( 𝜑 → ( ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ↔ ( 𝐺0𝐻 = 0 ) ) )

Proof

Step Hyp Ref Expression
1 lkrpss.f 𝐹 = ( LFnl ‘ 𝑊 )
2 lkrpss.k 𝐾 = ( LKer ‘ 𝑊 )
3 lkrpss.d 𝐷 = ( LDual ‘ 𝑊 )
4 lkrpss.o 0 = ( 0g𝐷 )
5 lkrpss.w ( 𝜑𝑊 ∈ LVec )
6 lkrpss.g ( 𝜑𝐺𝐹 )
7 lkrpss.h ( 𝜑𝐻𝐹 )
8 df-pss ( ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ↔ ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) )
9 simpr ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ) → ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) )
10 eqid ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 )
11 lveclmod ( 𝑊 ∈ LVec → 𝑊 ∈ LMod )
12 5 11 syl ( 𝜑𝑊 ∈ LMod )
13 10 1 2 12 7 lkrssv ( 𝜑 → ( 𝐾𝐻 ) ⊆ ( Base ‘ 𝑊 ) )
14 13 adantr ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ) → ( 𝐾𝐻 ) ⊆ ( Base ‘ 𝑊 ) )
15 9 14 psssstrd ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ) → ( 𝐾𝐺 ) ⊊ ( Base ‘ 𝑊 ) )
16 15 pssned ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ) → ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) )
17 8 16 sylan2br ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) ) → ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) )
18 simplr ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) )
19 eqid ( LSHyp ‘ 𝑊 ) = ( LSHyp ‘ 𝑊 )
20 5 ad2antrr ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → 𝑊 ∈ LVec )
21 simpr ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) )
22 simplr ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) )
23 13 ad3antrrr ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾𝐻 ) ⊆ ( Base ‘ 𝑊 ) )
24 simpr ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) )
25 simpllr ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) )
26 24 25 eqsstrrd ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( Base ‘ 𝑊 ) ⊆ ( 𝐾𝐻 ) )
27 23 26 eqssd ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) )
28 10 19 1 2 5 7 lkrshp4 ( 𝜑 → ( ( 𝐾𝐻 ) ≠ ( Base ‘ 𝑊 ) ↔ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) )
29 28 ad3antrrr ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( ( 𝐾𝐻 ) ≠ ( Base ‘ 𝑊 ) ↔ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) )
30 29 necon1bbid ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( ¬ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ↔ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) )
31 27 30 mpbird ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ¬ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) )
32 22 31 pm2.21dd ( ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) ∧ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) → ( 𝐾𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) )
33 10 19 1 2 5 6 lkrshpor ( 𝜑 → ( ( 𝐾𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ∨ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) )
34 33 ad2antrr ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( ( 𝐾𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) ∨ ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ) )
35 21 32 34 mpjaodan ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾𝐺 ) ∈ ( LSHyp ‘ 𝑊 ) )
36 simpr ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) )
37 19 20 35 36 lshpcmp ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ↔ ( 𝐾𝐺 ) = ( 𝐾𝐻 ) ) )
38 18 37 mpbid ( ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) ∧ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) → ( 𝐾𝐺 ) = ( 𝐾𝐻 ) )
39 38 ex ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) → ( ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) → ( 𝐾𝐺 ) = ( 𝐾𝐻 ) ) )
40 39 necon3ad ( ( 𝜑 ∧ ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ) → ( ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) → ¬ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ) )
41 40 impr ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) ) → ¬ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) )
42 28 necon1bbid ( 𝜑 → ( ¬ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ↔ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) )
43 42 adantr ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) ) → ( ¬ ( 𝐾𝐻 ) ∈ ( LSHyp ‘ 𝑊 ) ↔ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) )
44 41 43 mpbid ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) ) → ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) )
45 17 44 jca ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) ) → ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) )
46 10 1 2 12 6 lkrssv ( 𝜑 → ( 𝐾𝐺 ) ⊆ ( Base ‘ 𝑊 ) )
47 46 adantr ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾𝐺 ) ⊆ ( Base ‘ 𝑊 ) )
48 simprr ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) )
49 48 eqcomd ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( Base ‘ 𝑊 ) = ( 𝐾𝐻 ) )
50 47 49 sseqtrd ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) )
51 simprl ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) )
52 51 49 neeqtrd ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) )
53 50 52 jca ( ( 𝜑 ∧ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) → ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) )
54 45 53 impbida ( 𝜑 → ( ( ( 𝐾𝐺 ) ⊆ ( 𝐾𝐻 ) ∧ ( 𝐾𝐺 ) ≠ ( 𝐾𝐻 ) ) ↔ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) )
55 8 54 syl5bb ( 𝜑 → ( ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ↔ ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ) )
56 10 1 2 3 4 12 6 lkr0f2 ( 𝜑 → ( ( 𝐾𝐺 ) = ( Base ‘ 𝑊 ) ↔ 𝐺 = 0 ) )
57 56 necon3bid ( 𝜑 → ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ↔ 𝐺0 ) )
58 10 1 2 3 4 12 7 lkr0f2 ( 𝜑 → ( ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ↔ 𝐻 = 0 ) )
59 57 58 anbi12d ( 𝜑 → ( ( ( 𝐾𝐺 ) ≠ ( Base ‘ 𝑊 ) ∧ ( 𝐾𝐻 ) = ( Base ‘ 𝑊 ) ) ↔ ( 𝐺0𝐻 = 0 ) ) )
60 55 59 bitrd ( 𝜑 → ( ( 𝐾𝐺 ) ⊊ ( 𝐾𝐻 ) ↔ ( 𝐺0𝐻 = 0 ) ) )