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Theorem hhsssh 30777
Description: The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
hhsst.1 π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©
hhsst.2 π‘Š = ⟨⟨( +β„Ž β†Ύ (𝐻 Γ— 𝐻)), ( Β·β„Ž β†Ύ (β„‚ Γ— 𝐻))⟩, (normβ„Ž β†Ύ 𝐻)⟩
Assertion
Ref Expression
hhsssh (𝐻 ∈ Sβ„‹ ↔ (π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹))
Proof of Theorem hhsssh
StepHypRef Expression
1 hhsst.1 . . . 4 π‘ˆ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©
2 hhsst.2 . . . 4 π‘Š = ⟨⟨( +β„Ž β†Ύ (𝐻 Γ— 𝐻)), ( Β·β„Ž β†Ύ (β„‚ Γ— 𝐻))⟩, (normβ„Ž β†Ύ 𝐻)⟩
31, 2hhsst 30774 . . 3 (𝐻 ∈ Sβ„‹ β†’ π‘Š ∈ (SubSpβ€˜π‘ˆ))
4 shss 30718 . . 3 (𝐻 ∈ Sβ„‹ β†’ 𝐻 βŠ† β„‹)
53, 4jca 512 . 2 (𝐻 ∈ Sβ„‹ β†’ (π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹))
6 eleq1 2821 . . 3 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (𝐻 ∈ Sβ„‹ ↔ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) ∈ Sβ„‹ ))
7 eqid 2732 . . . 4 ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩ = ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩
8 xpeq1 5690 . . . . . . . . . . . . 13 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (𝐻 Γ— 𝐻) = (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— 𝐻))
9 xpeq2 5697 . . . . . . . . . . . . 13 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— 𝐻) = (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
108, 9eqtrd 2772 . . . . . . . . . . . 12 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (𝐻 Γ— 𝐻) = (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
1110reseq2d 5981 . . . . . . . . . . 11 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ( +β„Ž β†Ύ (𝐻 Γ— 𝐻)) = ( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))))
12 xpeq2 5697 . . . . . . . . . . . 12 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (β„‚ Γ— 𝐻) = (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
1312reseq2d 5981 . . . . . . . . . . 11 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ( Β·β„Ž β†Ύ (β„‚ Γ— 𝐻)) = ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))))
1411, 13opeq12d 4881 . . . . . . . . . 10 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ⟨( +β„Ž β†Ύ (𝐻 Γ— 𝐻)), ( Β·β„Ž β†Ύ (β„‚ Γ— 𝐻))⟩ = ⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩)
15 reseq2 5976 . . . . . . . . . 10 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (normβ„Ž β†Ύ 𝐻) = (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
1614, 15opeq12d 4881 . . . . . . . . 9 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ⟨⟨( +β„Ž β†Ύ (𝐻 Γ— 𝐻)), ( Β·β„Ž β†Ύ (β„‚ Γ— 𝐻))⟩, (normβ„Ž β†Ύ 𝐻)⟩ = ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩)
172, 16eqtrid 2784 . . . . . . . 8 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ π‘Š = ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩)
1817eleq1d 2818 . . . . . . 7 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (π‘Š ∈ (SubSpβ€˜π‘ˆ) ↔ ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩ ∈ (SubSpβ€˜π‘ˆ)))
19 sseq1 4007 . . . . . . 7 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (𝐻 βŠ† β„‹ ↔ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) βŠ† β„‹))
2018, 19anbi12d 631 . . . . . 6 (𝐻 = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹) ↔ (⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩ ∈ (SubSpβ€˜π‘ˆ) ∧ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) βŠ† β„‹)))
21 xpeq1 5690 . . . . . . . . . . . 12 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ( β„‹ Γ— β„‹) = (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— β„‹))
22 xpeq2 5697 . . . . . . . . . . . 12 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— β„‹) = (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
2321, 22eqtrd 2772 . . . . . . . . . . 11 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ( β„‹ Γ— β„‹) = (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
2423reseq2d 5981 . . . . . . . . . 10 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ( +β„Ž β†Ύ ( β„‹ Γ— β„‹)) = ( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))))
25 xpeq2 5697 . . . . . . . . . . 11 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (β„‚ Γ— β„‹) = (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
2625reseq2d 5981 . . . . . . . . . 10 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹)) = ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))))
2724, 26opeq12d 4881 . . . . . . . . 9 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩ = ⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩)
28 reseq2 5976 . . . . . . . . 9 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (normβ„Ž β†Ύ β„‹) = (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))
2927, 28opeq12d 4881 . . . . . . . 8 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ = ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩)
3029eleq1d 2818 . . . . . . 7 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ (⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ ∈ (SubSpβ€˜π‘ˆ) ↔ ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩ ∈ (SubSpβ€˜π‘ˆ)))
31 sseq1 4007 . . . . . . 7 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ( β„‹ βŠ† β„‹ ↔ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) βŠ† β„‹))
3230, 31anbi12d 631 . . . . . 6 ( β„‹ = if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) β†’ ((⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ ∈ (SubSpβ€˜π‘ˆ) ∧ β„‹ βŠ† β„‹) ↔ (⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩ ∈ (SubSpβ€˜π‘ˆ) ∧ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) βŠ† β„‹)))
33 ax-hfvadd 30508 . . . . . . . . . . . 12 +β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
34 ffn 6717 . . . . . . . . . . . 12 ( +β„Ž :( β„‹ Γ— β„‹)⟢ β„‹ β†’ +β„Ž Fn ( β„‹ Γ— β„‹))
35 fnresdm 6669 . . . . . . . . . . . 12 ( +β„Ž Fn ( β„‹ Γ— β„‹) β†’ ( +β„Ž β†Ύ ( β„‹ Γ— β„‹)) = +β„Ž )
3633, 34, 35mp2b 10 . . . . . . . . . . 11 ( +β„Ž β†Ύ ( β„‹ Γ— β„‹)) = +β„Ž
37 ax-hfvmul 30513 . . . . . . . . . . . 12 Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹
38 ffn 6717 . . . . . . . . . . . 12 ( Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹ β†’ Β·β„Ž Fn (β„‚ Γ— β„‹))
39 fnresdm 6669 . . . . . . . . . . . 12 ( Β·β„Ž Fn (β„‚ Γ— β„‹) β†’ ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹)) = Β·β„Ž )
4037, 38, 39mp2b 10 . . . . . . . . . . 11 ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹)) = Β·β„Ž
4136, 40opeq12i 4878 . . . . . . . . . 10 ⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩ = ⟨ +β„Ž , Β·β„Ž ⟩
42 normf 30631 . . . . . . . . . . 11 normβ„Ž: β„‹βŸΆβ„
43 ffn 6717 . . . . . . . . . . 11 (normβ„Ž: β„‹βŸΆβ„ β†’ normβ„Ž Fn β„‹)
44 fnresdm 6669 . . . . . . . . . . 11 (normβ„Ž Fn β„‹ β†’ (normβ„Ž β†Ύ β„‹) = normβ„Ž)
4542, 43, 44mp2b 10 . . . . . . . . . 10 (normβ„Ž β†Ύ β„‹) = normβ„Ž
4641, 45opeq12i 4878 . . . . . . . . 9 ⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ = ⟨⟨ +β„Ž , Β·β„Ž ⟩, normβ„ŽβŸ©
4746, 1eqtr4i 2763 . . . . . . . 8 ⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ = π‘ˆ
481hhnv 30673 . . . . . . . . 9 π‘ˆ ∈ NrmCVec
49 eqid 2732 . . . . . . . . . 10 (SubSpβ€˜π‘ˆ) = (SubSpβ€˜π‘ˆ)
5049sspid 30233 . . . . . . . . 9 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ ∈ (SubSpβ€˜π‘ˆ))
5148, 50ax-mp 5 . . . . . . . 8 π‘ˆ ∈ (SubSpβ€˜π‘ˆ)
5247, 51eqeltri 2829 . . . . . . 7 ⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ ∈ (SubSpβ€˜π‘ˆ)
53 ssid 4004 . . . . . . 7 β„‹ βŠ† β„‹
5452, 53pm3.2i 471 . . . . . 6 (⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ ∈ (SubSpβ€˜π‘ˆ) ∧ β„‹ βŠ† β„‹)
5520, 32, 54elimhyp 4593 . . . . 5 (⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩ ∈ (SubSpβ€˜π‘ˆ) ∧ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) βŠ† β„‹)
5655simpli 484 . . . 4 ⟨⟨( +β„Ž β†Ύ (if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))), ( Β·β„Ž β†Ύ (β„‚ Γ— if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹)))⟩, (normβ„Ž β†Ύ if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹))⟩ ∈ (SubSpβ€˜π‘ˆ)
5755simpri 486 . . . 4 if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) βŠ† β„‹
581, 7, 56, 57hhshsslem2 30776 . . 3 if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) ∈ Sβ„‹
596, 58dedth 4586 . 2 ((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹) β†’ 𝐻 ∈ Sβ„‹ )
605, 59impbii 208 1 (𝐻 ∈ Sβ„‹ ↔ (π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  ifcif 4528  βŸ¨cop 4634   Γ— cxp 5674   β†Ύ cres 5678   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  β„‚cc 11110  β„cr 11111  NrmCVeccnv 30092  SubSpcss 30229   β„‹chba 30427   +β„Ž cva 30428   Β·β„Ž csm 30429  normβ„Žcno 30431   Sβ„‹ csh 30436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192  ax-hilex 30507  ax-hfvadd 30508  ax-hvcom 30509  ax-hvass 30510  ax-hv0cl 30511  ax-hvaddid 30512  ax-hfvmul 30513  ax-hvmulid 30514  ax-hvmulass 30515  ax-hvdistr1 30516  ax-hvdistr2 30517  ax-hvmul0 30518  ax-hfi 30587  ax-his1 30590  ax-his2 30591  ax-his3 30592  ax-his4 30593
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-icc 13335  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-topgen 17393  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-top 22616  df-topon 22633  df-bases 22669  df-lm 22953  df-haus 23039  df-grpo 30001  df-gid 30002  df-ginv 30003  df-gdiv 30004  df-ablo 30053  df-vc 30067  df-nv 30100  df-va 30103  df-ba 30104  df-sm 30105  df-0v 30106  df-vs 30107  df-nmcv 30108  df-ims 30109  df-ssp 30230  df-hnorm 30476  df-hba 30477  df-hvsub 30479  df-hlim 30480  df-sh 30715  df-ch 30729  df-ch0 30761
This theorem is referenced by:  hhsssh2  30778
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