Metamath Proof Explorer


Theorem sitgaddlemb

Description: Lemma for * sitgadd . (Contributed by Thierry Arnoux, 10-Mar-2019)

Ref Expression
Hypotheses sitgval.b 𝐵 = ( Base ‘ 𝑊 )
sitgval.j 𝐽 = ( TopOpen ‘ 𝑊 )
sitgval.s 𝑆 = ( sigaGen ‘ 𝐽 )
sitgval.0 0 = ( 0g𝑊 )
sitgval.x · = ( ·𝑠𝑊 )
sitgval.h 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
sitgval.1 ( 𝜑𝑊𝑉 )
sitgval.2 ( 𝜑𝑀 ran measures )
sitgadd.1 ( 𝜑𝑊 ∈ TopSp )
sitgadd.2 ( 𝜑 → ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ∈ SLMod )
sitgadd.3 ( 𝜑𝐽 ∈ Fre )
sitgadd.4 ( 𝜑𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) )
sitgadd.5 ( 𝜑𝐺 ∈ dom ( 𝑊 sitg 𝑀 ) )
sitgadd.6 ( 𝜑 → ( Scalar ‘ 𝑊 ) ∈ ℝExt )
sitgadd.7 + = ( +g𝑊 )
Assertion sitgaddlemb ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( ( 𝐻 ‘ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ) · ( 2nd𝑝 ) ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 sitgval.b 𝐵 = ( Base ‘ 𝑊 )
2 sitgval.j 𝐽 = ( TopOpen ‘ 𝑊 )
3 sitgval.s 𝑆 = ( sigaGen ‘ 𝐽 )
4 sitgval.0 0 = ( 0g𝑊 )
5 sitgval.x · = ( ·𝑠𝑊 )
6 sitgval.h 𝐻 = ( ℝHom ‘ ( Scalar ‘ 𝑊 ) )
7 sitgval.1 ( 𝜑𝑊𝑉 )
8 sitgval.2 ( 𝜑𝑀 ran measures )
9 sitgadd.1 ( 𝜑𝑊 ∈ TopSp )
10 sitgadd.2 ( 𝜑 → ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ∈ SLMod )
11 sitgadd.3 ( 𝜑𝐽 ∈ Fre )
12 sitgadd.4 ( 𝜑𝐹 ∈ dom ( 𝑊 sitg 𝑀 ) )
13 sitgadd.5 ( 𝜑𝐺 ∈ dom ( 𝑊 sitg 𝑀 ) )
14 sitgadd.6 ( 𝜑 → ( Scalar ‘ 𝑊 ) ∈ ℝExt )
15 sitgadd.7 + = ( +g𝑊 )
16 10 adantr ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ∈ SLMod )
17 simpl ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → 𝜑 )
18 eqid ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
19 18 rrhfe ( ( Scalar ‘ 𝑊 ) ∈ ℝExt → ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
20 14 19 syl ( 𝜑 → ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
21 6 feq1i ( 𝐻 : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ↔ ( ℝHom ‘ ( Scalar ‘ 𝑊 ) ) : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
22 20 21 sylibr ( 𝜑𝐻 : ℝ ⟶ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
23 22 ffnd ( 𝜑𝐻 Fn ℝ )
24 17 23 syl ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → 𝐻 Fn ℝ )
25 rge0ssre ( 0 [,) +∞ ) ⊆ ℝ
26 25 a1i ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 0 [,) +∞ ) ⊆ ℝ )
27 simpr ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → 𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) )
28 27 eldifad ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) )
29 xp1st ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( 1st𝑝 ) ∈ ran 𝐹 )
30 28 29 syl ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 1st𝑝 ) ∈ ran 𝐹 )
31 xp2nd ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( 2nd𝑝 ) ∈ ran 𝐺 )
32 28 31 syl ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 2nd𝑝 ) ∈ ran 𝐺 )
33 27 eldifbd ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ¬ 𝑝 ∈ { ⟨ 0 , 0 ⟩ } )
34 velsn ( 𝑝 ∈ { ⟨ 0 , 0 ⟩ } ↔ 𝑝 = ⟨ 0 , 0 ⟩ )
35 34 notbii ( ¬ 𝑝 ∈ { ⟨ 0 , 0 ⟩ } ↔ ¬ 𝑝 = ⟨ 0 , 0 ⟩ )
36 33 35 sylib ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ¬ 𝑝 = ⟨ 0 , 0 ⟩ )
37 eqopi ( ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) ∧ ( ( 1st𝑝 ) = 0 ∧ ( 2nd𝑝 ) = 0 ) ) → 𝑝 = ⟨ 0 , 0 ⟩ )
38 37 ex ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( ( ( 1st𝑝 ) = 0 ∧ ( 2nd𝑝 ) = 0 ) → 𝑝 = ⟨ 0 , 0 ⟩ ) )
39 38 con3d ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) → ( ¬ 𝑝 = ⟨ 0 , 0 ⟩ → ¬ ( ( 1st𝑝 ) = 0 ∧ ( 2nd𝑝 ) = 0 ) ) )
40 39 imp ( ( 𝑝 ∈ ( ran 𝐹 × ran 𝐺 ) ∧ ¬ 𝑝 = ⟨ 0 , 0 ⟩ ) → ¬ ( ( 1st𝑝 ) = 0 ∧ ( 2nd𝑝 ) = 0 ) )
41 28 36 40 syl2anc ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ¬ ( ( 1st𝑝 ) = 0 ∧ ( 2nd𝑝 ) = 0 ) )
42 ianor ( ¬ ( ( 1st𝑝 ) = 0 ∧ ( 2nd𝑝 ) = 0 ) ↔ ( ¬ ( 1st𝑝 ) = 0 ∨ ¬ ( 2nd𝑝 ) = 0 ) )
43 df-ne ( ( 1st𝑝 ) ≠ 0 ↔ ¬ ( 1st𝑝 ) = 0 )
44 df-ne ( ( 2nd𝑝 ) ≠ 0 ↔ ¬ ( 2nd𝑝 ) = 0 )
45 43 44 orbi12i ( ( ( 1st𝑝 ) ≠ 0 ∨ ( 2nd𝑝 ) ≠ 0 ) ↔ ( ¬ ( 1st𝑝 ) = 0 ∨ ¬ ( 2nd𝑝 ) = 0 ) )
46 42 45 bitr4i ( ¬ ( ( 1st𝑝 ) = 0 ∧ ( 2nd𝑝 ) = 0 ) ↔ ( ( 1st𝑝 ) ≠ 0 ∨ ( 2nd𝑝 ) ≠ 0 ) )
47 41 46 sylib ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( ( 1st𝑝 ) ≠ 0 ∨ ( 2nd𝑝 ) ≠ 0 ) )
48 1 2 3 4 5 6 7 8 12 13 9 11 sibfinima ( ( ( 𝜑 ∧ ( 1st𝑝 ) ∈ ran 𝐹 ∧ ( 2nd𝑝 ) ∈ ran 𝐺 ) ∧ ( ( 1st𝑝 ) ≠ 0 ∨ ( 2nd𝑝 ) ≠ 0 ) ) → ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ∈ ( 0 [,) +∞ ) )
49 17 30 32 47 48 syl31anc ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ∈ ( 0 [,) +∞ ) )
50 fnfvima ( ( 𝐻 Fn ℝ ∧ ( 0 [,) +∞ ) ⊆ ℝ ∧ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ∈ ( 0 [,) +∞ ) ) → ( 𝐻 ‘ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ) ∈ ( 𝐻 “ ( 0 [,) +∞ ) ) )
51 24 26 49 50 syl3anc ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 𝐻 ‘ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ) ∈ ( 𝐻 “ ( 0 [,) +∞ ) ) )
52 imassrn ( 𝐻 “ ( 0 [,) +∞ ) ) ⊆ ran 𝐻
53 22 frnd ( 𝜑 → ran 𝐻 ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
54 52 53 sstrid ( 𝜑 → ( 𝐻 “ ( 0 [,) +∞ ) ) ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
55 eqid ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) )
56 55 18 ressbas2 ( ( 𝐻 “ ( 0 [,) +∞ ) ) ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) → ( 𝐻 “ ( 0 [,) +∞ ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) )
57 54 56 syl ( 𝜑 → ( 𝐻 “ ( 0 [,) +∞ ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) )
58 17 57 syl ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 𝐻 “ ( 0 [,) +∞ ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) )
59 51 58 eleqtrd ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 𝐻 ‘ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ) ∈ ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) )
60 1 2 3 4 5 6 7 8 13 sibff ( 𝜑𝐺 : dom 𝑀 𝐽 )
61 1 2 tpsuni ( 𝑊 ∈ TopSp → 𝐵 = 𝐽 )
62 feq3 ( 𝐵 = 𝐽 → ( 𝐺 : dom 𝑀𝐵𝐺 : dom 𝑀 𝐽 ) )
63 9 61 62 3syl ( 𝜑 → ( 𝐺 : dom 𝑀𝐵𝐺 : dom 𝑀 𝐽 ) )
64 60 63 mpbird ( 𝜑𝐺 : dom 𝑀𝐵 )
65 64 frnd ( 𝜑 → ran 𝐺𝐵 )
66 65 adantr ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ran 𝐺𝐵 )
67 66 32 sseldd ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( 2nd𝑝 ) ∈ 𝐵 )
68 6 fvexi 𝐻 ∈ V
69 imaexg ( 𝐻 ∈ V → ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V )
70 eqid ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) )
71 70 1 resvbas ( ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V → 𝐵 = ( Base ‘ ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) )
72 68 69 71 mp2b 𝐵 = ( Base ‘ ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) )
73 eqid ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
74 70 73 18 resvsca ( ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V → ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( Scalar ‘ ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) )
75 68 69 74 mp2b ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) = ( Scalar ‘ ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) )
76 70 5 resvvsca ( ( 𝐻 “ ( 0 [,) +∞ ) ) ∈ V → · = ( ·𝑠 ‘ ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) )
77 68 69 76 mp2b · = ( ·𝑠 ‘ ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) )
78 eqid ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) = ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) )
79 72 75 77 78 slmdvscl ( ( ( 𝑊v ( 𝐻 “ ( 0 [,) +∞ ) ) ) ∈ SLMod ∧ ( 𝐻 ‘ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ) ∈ ( Base ‘ ( ( Scalar ‘ 𝑊 ) ↾s ( 𝐻 “ ( 0 [,) +∞ ) ) ) ) ∧ ( 2nd𝑝 ) ∈ 𝐵 ) → ( ( 𝐻 ‘ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ) · ( 2nd𝑝 ) ) ∈ 𝐵 )
80 16 59 67 79 syl3anc ( ( 𝜑𝑝 ∈ ( ( ran 𝐹 × ran 𝐺 ) ∖ { ⟨ 0 , 0 ⟩ } ) ) → ( ( 𝐻 ‘ ( 𝑀 ‘ ( ( 𝐹 “ { ( 1st𝑝 ) } ) ∩ ( 𝐺 “ { ( 2nd𝑝 ) } ) ) ) ) · ( 2nd𝑝 ) ) ∈ 𝐵 )