Metamath Proof Explorer


Theorem dia2dimlem9

Description: Lemma for dia2dim . Eliminate ( RF ) =/= U , V conditions. (Contributed by NM, 8-Sep-2014)

Ref Expression
Hypotheses dia2dimlem9.l = ( le ‘ 𝐾 )
dia2dimlem9.j = ( join ‘ 𝐾 )
dia2dimlem9.m = ( meet ‘ 𝐾 )
dia2dimlem9.a 𝐴 = ( Atoms ‘ 𝐾 )
dia2dimlem9.h 𝐻 = ( LHyp ‘ 𝐾 )
dia2dimlem9.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dia2dimlem9.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
dia2dimlem9.y 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
dia2dimlem9.s 𝑆 = ( LSubSp ‘ 𝑌 )
dia2dimlem9.pl = ( LSSum ‘ 𝑌 )
dia2dimlem9.n 𝑁 = ( LSpan ‘ 𝑌 )
dia2dimlem9.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
dia2dimlem9.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
dia2dimlem9.u ( 𝜑 → ( 𝑈𝐴𝑈 𝑊 ) )
dia2dimlem9.v ( 𝜑 → ( 𝑉𝐴𝑉 𝑊 ) )
dia2dimlem9.f ( 𝜑𝐹𝑇 )
dia2dimlem9.rf ( 𝜑 → ( 𝑅𝐹 ) ( 𝑈 𝑉 ) )
dia2dimlem9.uv ( 𝜑𝑈𝑉 )
Assertion dia2dimlem9 ( 𝜑𝐹 ∈ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )

Proof

Step Hyp Ref Expression
1 dia2dimlem9.l = ( le ‘ 𝐾 )
2 dia2dimlem9.j = ( join ‘ 𝐾 )
3 dia2dimlem9.m = ( meet ‘ 𝐾 )
4 dia2dimlem9.a 𝐴 = ( Atoms ‘ 𝐾 )
5 dia2dimlem9.h 𝐻 = ( LHyp ‘ 𝐾 )
6 dia2dimlem9.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 dia2dimlem9.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 dia2dimlem9.y 𝑌 = ( ( DVecA ‘ 𝐾 ) ‘ 𝑊 )
9 dia2dimlem9.s 𝑆 = ( LSubSp ‘ 𝑌 )
10 dia2dimlem9.pl = ( LSSum ‘ 𝑌 )
11 dia2dimlem9.n 𝑁 = ( LSpan ‘ 𝑌 )
12 dia2dimlem9.i 𝐼 = ( ( DIsoA ‘ 𝐾 ) ‘ 𝑊 )
13 dia2dimlem9.k ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
14 dia2dimlem9.u ( 𝜑 → ( 𝑈𝐴𝑈 𝑊 ) )
15 dia2dimlem9.v ( 𝜑 → ( 𝑉𝐴𝑉 𝑊 ) )
16 dia2dimlem9.f ( 𝜑𝐹𝑇 )
17 dia2dimlem9.rf ( 𝜑 → ( 𝑅𝐹 ) ( 𝑈 𝑉 ) )
18 dia2dimlem9.uv ( 𝜑𝑈𝑉 )
19 5 8 dvalvec ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝑌 ∈ LVec )
20 lveclmod ( 𝑌 ∈ LVec → 𝑌 ∈ LMod )
21 9 lsssssubg ( 𝑌 ∈ LMod → 𝑆 ⊆ ( SubGrp ‘ 𝑌 ) )
22 13 19 20 21 4syl ( 𝜑𝑆 ⊆ ( SubGrp ‘ 𝑌 ) )
23 14 simpld ( 𝜑𝑈𝐴 )
24 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
25 24 4 atbase ( 𝑈𝐴𝑈 ∈ ( Base ‘ 𝐾 ) )
26 23 25 syl ( 𝜑𝑈 ∈ ( Base ‘ 𝐾 ) )
27 14 simprd ( 𝜑𝑈 𝑊 )
28 24 1 5 8 12 9 dialss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 𝑊 ) ) → ( 𝐼𝑈 ) ∈ 𝑆 )
29 13 26 27 28 syl12anc ( 𝜑 → ( 𝐼𝑈 ) ∈ 𝑆 )
30 22 29 sseldd ( 𝜑 → ( 𝐼𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) )
31 15 simpld ( 𝜑𝑉𝐴 )
32 24 4 atbase ( 𝑉𝐴𝑉 ∈ ( Base ‘ 𝐾 ) )
33 31 32 syl ( 𝜑𝑉 ∈ ( Base ‘ 𝐾 ) )
34 15 simprd ( 𝜑𝑉 𝑊 )
35 24 1 5 8 12 9 dialss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 𝑊 ) ) → ( 𝐼𝑉 ) ∈ 𝑆 )
36 13 33 34 35 syl12anc ( 𝜑 → ( 𝐼𝑉 ) ∈ 𝑆 )
37 22 36 sseldd ( 𝜑 → ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) )
38 10 lsmub1 ( ( ( 𝐼𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ∧ ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ) → ( 𝐼𝑈 ) ⊆ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
39 30 37 38 syl2anc ( 𝜑 → ( 𝐼𝑈 ) ⊆ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
40 39 adantr ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑈 ) → ( 𝐼𝑈 ) ⊆ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
41 5 6 7 12 dia1dimid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ) → 𝐹 ∈ ( 𝐼 ‘ ( 𝑅𝐹 ) ) )
42 13 16 41 syl2anc ( 𝜑𝐹 ∈ ( 𝐼 ‘ ( 𝑅𝐹 ) ) )
43 42 adantr ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑈 ) → 𝐹 ∈ ( 𝐼 ‘ ( 𝑅𝐹 ) ) )
44 fveq2 ( ( 𝑅𝐹 ) = 𝑈 → ( 𝐼 ‘ ( 𝑅𝐹 ) ) = ( 𝐼𝑈 ) )
45 44 adantl ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑈 ) → ( 𝐼 ‘ ( 𝑅𝐹 ) ) = ( 𝐼𝑈 ) )
46 43 45 eleqtrd ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑈 ) → 𝐹 ∈ ( 𝐼𝑈 ) )
47 40 46 sseldd ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑈 ) → 𝐹 ∈ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
48 30 adantr ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑉 ) → ( 𝐼𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) )
49 37 adantr ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑉 ) → ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) )
50 10 lsmub2 ( ( ( 𝐼𝑈 ) ∈ ( SubGrp ‘ 𝑌 ) ∧ ( 𝐼𝑉 ) ∈ ( SubGrp ‘ 𝑌 ) ) → ( 𝐼𝑉 ) ⊆ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
51 48 49 50 syl2anc ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑉 ) → ( 𝐼𝑉 ) ⊆ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
52 42 adantr ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑉 ) → 𝐹 ∈ ( 𝐼 ‘ ( 𝑅𝐹 ) ) )
53 fveq2 ( ( 𝑅𝐹 ) = 𝑉 → ( 𝐼 ‘ ( 𝑅𝐹 ) ) = ( 𝐼𝑉 ) )
54 53 adantl ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑉 ) → ( 𝐼 ‘ ( 𝑅𝐹 ) ) = ( 𝐼𝑉 ) )
55 52 54 eleqtrd ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑉 ) → 𝐹 ∈ ( 𝐼𝑉 ) )
56 51 55 sseldd ( ( 𝜑 ∧ ( 𝑅𝐹 ) = 𝑉 ) → 𝐹 ∈ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
57 13 adantr ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
58 14 adantr ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → ( 𝑈𝐴𝑈 𝑊 ) )
59 15 adantr ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → ( 𝑉𝐴𝑉 𝑊 ) )
60 16 adantr ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → 𝐹𝑇 )
61 17 adantr ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → ( 𝑅𝐹 ) ( 𝑈 𝑉 ) )
62 18 adantr ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → 𝑈𝑉 )
63 simprl ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → ( 𝑅𝐹 ) ≠ 𝑈 )
64 simprr ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → ( 𝑅𝐹 ) ≠ 𝑉 )
65 1 2 3 4 5 6 7 8 9 10 11 12 57 58 59 60 61 62 63 64 dia2dimlem8 ( ( 𝜑 ∧ ( ( 𝑅𝐹 ) ≠ 𝑈 ∧ ( 𝑅𝐹 ) ≠ 𝑉 ) ) → 𝐹 ∈ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )
66 47 56 65 pm2.61da2ne ( 𝜑𝐹 ∈ ( ( 𝐼𝑈 ) ( 𝐼𝑉 ) ) )