Metamath Proof Explorer

Theorem dihf11lem

Description: Functionality of the isomorphism H. (Contributed by NM, 6-Mar-2014)

Ref Expression
Hypotheses dihf11.b 𝐵 = ( Base ‘ 𝐾 )
dihf11.h 𝐻 = ( LHyp ‘ 𝐾 )
dihf11.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
dihf11.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihf11.s 𝑆 = ( LSubSp ‘ 𝑈 )
Assertion dihf11lem ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐼 : 𝐵𝑆 )

Proof

Step Hyp Ref Expression
1 dihf11.b 𝐵 = ( Base ‘ 𝐾 )
2 dihf11.h 𝐻 = ( LHyp ‘ 𝐾 )
3 dihf11.i 𝐼 = ( ( DIsoH ‘ 𝐾 ) ‘ 𝑊 )
4 dihf11.u 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5 dihf11.s 𝑆 = ( LSubSp ‘ 𝑈 )
6 fvex ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) ∈ V
7 riotaex ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ∈ V
8 6 7 ifex if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ∈ V
9 8 rgenw 𝑥𝐵 if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ∈ V
10 9 a1i ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ∀ 𝑥𝐵 if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ∈ V )
11 eqid ( 𝑥𝐵 ↦ if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ) = ( 𝑥𝐵 ↦ if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) )
12 11 mptfng ( ∀ 𝑥𝐵 if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ∈ V ↔ ( 𝑥𝐵 ↦ if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ) Fn 𝐵 )
13 10 12 sylib ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝑥𝐵 ↦ if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ) Fn 𝐵 )
14 eqid ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
15 eqid ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
16 eqid ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
17 eqid ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
18 eqid ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
19 eqid ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
20 eqid ( LSSum ‘ 𝑈 ) = ( LSSum ‘ 𝑈 )
21 1 14 15 16 17 2 3 18 19 4 5 20 dihfval ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐼 = ( 𝑥𝐵 ↦ if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ) )
22 21 fneq1d ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝐼 Fn 𝐵 ↔ ( 𝑥𝐵 ↦ if ( 𝑥 ( le ‘ 𝐾 ) 𝑊 , ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑥 ) , ( 𝑢𝑆𝑞 ∈ ( Atoms ‘ 𝐾 ) ( ( ¬ 𝑞 ( le ‘ 𝐾 ) 𝑊 ∧ ( 𝑞 ( join ‘ 𝐾 ) ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) = 𝑥 ) → 𝑢 = ( ( ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝑞 ) ( LSSum ‘ 𝑈 ) ( ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 ) ‘ ( 𝑥 ( meet ‘ 𝐾 ) 𝑊 ) ) ) ) ) ) ) Fn 𝐵 ) )
23 13 22 mpbird ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐼 Fn 𝐵 )
24 1 2 3 4 5 dihlss ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑦𝐵 ) → ( 𝐼𝑦 ) ∈ 𝑆 )
25 24 ralrimiva ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ∀ 𝑦𝐵 ( 𝐼𝑦 ) ∈ 𝑆 )
26 fnfvrnss ( ( 𝐼 Fn 𝐵 ∧ ∀ 𝑦𝐵 ( 𝐼𝑦 ) ∈ 𝑆 ) → ran 𝐼𝑆 )
27 23 25 26 syl2anc ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ran 𝐼𝑆 )
28 df-f ( 𝐼 : 𝐵𝑆 ↔ ( 𝐼 Fn 𝐵 ∧ ran 𝐼𝑆 ) )
29 23 27 28 sylanbrc ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → 𝐼 : 𝐵𝑆 )