Metamath Proof Explorer


Theorem tendospcanN

Description: Cancellation law for trace-preserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses tendospcan.b 𝐵 = ( Base ‘ 𝐾 )
tendospcan.h 𝐻 = ( LHyp ‘ 𝐾 )
tendospcan.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
tendospcan.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
tendospcan.o 𝑂 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
Assertion tendospcanN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐸𝑆𝑂 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ↔ 𝐹 = 𝐺 ) )

Proof

Step Hyp Ref Expression
1 tendospcan.b 𝐵 = ( Base ‘ 𝐾 )
2 tendospcan.h 𝐻 = ( LHyp ‘ 𝐾 )
3 tendospcan.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 tendospcan.e 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5 tendospcan.o 𝑂 = ( 𝑓𝑇 ↦ ( I ↾ 𝐵 ) )
6 2 3 4 tendocnv ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸𝐺𝑇 ) → ( 𝑆𝐺 ) = ( 𝑆 𝐺 ) )
7 6 3adant3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝑆𝐺 ) = ( 𝑆 𝐺 ) )
8 7 coeq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆 𝐺 ) ) )
9 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simp2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝑆𝐸 )
11 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝐹𝑇 )
12 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝐺𝑇 )
13 2 3 ltrncnv ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇 ) → 𝐺𝑇 )
14 9 12 13 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝐺𝑇 )
15 2 3 4 tendospdi1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐸𝐹𝑇 𝐺𝑇 ) ) → ( 𝑆 ‘ ( 𝐹 𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆 𝐺 ) ) )
16 9 10 11 14 15 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝑆 ‘ ( 𝐹 𝐺 ) ) = ( ( 𝑆𝐹 ) ∘ ( 𝑆 𝐺 ) ) )
17 8 16 eqtr4d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) = ( 𝑆 ‘ ( 𝐹 𝐺 ) ) )
18 17 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) = ( 𝑆 ‘ ( 𝐹 𝐺 ) ) )
19 18 eqeq1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) = ( I ↾ 𝐵 ) ↔ ( 𝑆 ‘ ( 𝐹 𝐺 ) ) = ( I ↾ 𝐵 ) ) )
20 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
21 simpl2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → 𝑆𝐸 )
22 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → 𝐹𝑇 )
23 2 3 4 tendocl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸𝐹𝑇 ) → ( 𝑆𝐹 ) ∈ 𝑇 )
24 20 21 22 23 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝑆𝐹 ) ∈ 𝑇 )
25 simpl3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → 𝐺𝑇 )
26 2 3 4 tendocl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸𝐺𝑇 ) → ( 𝑆𝐺 ) ∈ 𝑇 )
27 20 21 25 26 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝑆𝐺 ) ∈ 𝑇 )
28 1 2 3 ltrncoidN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐹 ) ∈ 𝑇 ∧ ( 𝑆𝐺 ) ∈ 𝑇 ) → ( ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) = ( I ↾ 𝐵 ) ↔ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) )
29 20 24 27 28 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( ( 𝑆𝐹 ) ∘ ( 𝑆𝐺 ) ) = ( I ↾ 𝐵 ) ↔ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) )
30 20 25 13 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → 𝐺𝑇 )
31 2 3 ltrnco ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 𝐺𝑇 ) → ( 𝐹 𝐺 ) ∈ 𝑇 )
32 20 22 30 31 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝐹 𝐺 ) ∈ 𝑇 )
33 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) )
34 1 2 3 4 5 tendoid0 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( ( 𝐹 𝐺 ) ∈ 𝑇 ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) ) → ( ( 𝑆 ‘ ( 𝐹 𝐺 ) ) = ( I ↾ 𝐵 ) ↔ 𝑆 = 𝑂 ) )
35 20 21 32 33 34 syl112anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆 ‘ ( 𝐹 𝐺 ) ) = ( I ↾ 𝐵 ) ↔ 𝑆 = 𝑂 ) )
36 19 29 35 3bitr3d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ↔ 𝑆 = 𝑂 ) )
37 36 biimpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) → 𝑆 = 𝑂 ) )
38 37 impancom ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) → ( ( 𝐹 𝐺 ) ≠ ( I ↾ 𝐵 ) → 𝑆 = 𝑂 ) )
39 38 necon1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) → ( 𝑆𝑂 → ( 𝐹 𝐺 ) = ( I ↾ 𝐵 ) ) )
40 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
41 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) → 𝐹𝑇 )
42 simpl3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) → 𝐺𝑇 )
43 1 2 3 ltrncoidN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝐺𝑇 ) → ( ( 𝐹 𝐺 ) = ( I ↾ 𝐵 ) ↔ 𝐹 = 𝐺 ) )
44 40 41 42 43 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) → ( ( 𝐹 𝐺 ) = ( I ↾ 𝐵 ) ↔ 𝐹 = 𝐺 ) )
45 39 44 sylibd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑆𝐸 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ) → ( 𝑆𝑂𝐹 = 𝐺 ) )
46 45 3exp1 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( 𝑆𝐸 → ( ( 𝐹𝑇𝐺𝑇 ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) → ( 𝑆𝑂𝐹 = 𝐺 ) ) ) ) )
47 46 com24 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) → ( ( 𝐹𝑇𝐺𝑇 ) → ( 𝑆𝐸 → ( 𝑆𝑂𝐹 = 𝐺 ) ) ) ) )
48 47 imp5a ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) → ( ( 𝐹𝑇𝐺𝑇 ) → ( ( 𝑆𝐸𝑆𝑂 ) → 𝐹 = 𝐺 ) ) ) )
49 48 com24 ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( ( 𝑆𝐸𝑆𝑂 ) → ( ( 𝐹𝑇𝐺𝑇 ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) → 𝐹 = 𝐺 ) ) ) )
50 49 3imp ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐸𝑆𝑂 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) → 𝐹 = 𝐺 ) )
51 fveq2 ( 𝐹 = 𝐺 → ( 𝑆𝐹 ) = ( 𝑆𝐺 ) )
52 50 51 impbid1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐸𝑆𝑂 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑆𝐹 ) = ( 𝑆𝐺 ) ↔ 𝐹 = 𝐺 ) )