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Theorem hhsssh 30553
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 30550 . . 3 (𝐻 ∈ Sβ„‹ β†’ π‘Š ∈ (SubSpβ€˜π‘ˆ))
4 shss 30494 . . 3 (𝐻 ∈ Sβ„‹ β†’ 𝐻 βŠ† β„‹)
53, 4jca 513 . 2 (𝐻 ∈ Sβ„‹ β†’ (π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹))
6 eleq1 2822 . . 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 2733 . . . 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 5691 . . . . . . . . . . . . 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 5698 . . . . . . . . . . . . 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 2773 . . . . . . . . . . . 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 5982 . . . . . . . . . . 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 5698 . . . . . . . . . . . 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 5982 . . . . . . . . . . 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 4882 . . . . . . . . . 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 5977 . . . . . . . . . 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 4882 . . . . . . . . 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 2785 . . . . . . . 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 2819 . . . . . . 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 4008 . . . . . . 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 632 . . . . . 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 5691 . . . . . . . . . . . 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 5698 . . . . . . . . . . . 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 2773 . . . . . . . . . . 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 5982 . . . . . . . . . 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 5698 . . . . . . . . . . 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 5982 . . . . . . . . . 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 4882 . . . . . . . . 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 5977 . . . . . . . . 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 4882 . . . . . . . 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 2819 . . . . . . 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 4008 . . . . . . 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 632 . . . . . 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 30284 . . . . . . . . . . . 12 +β„Ž :( β„‹ Γ— β„‹)⟢ β„‹
34 ffn 6718 . . . . . . . . . . . 12 ( +β„Ž :( β„‹ Γ— β„‹)⟢ β„‹ β†’ +β„Ž Fn ( β„‹ Γ— β„‹))
35 fnresdm 6670 . . . . . . . . . . . 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 30289 . . . . . . . . . . . 12 Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹
38 ffn 6718 . . . . . . . . . . . 12 ( Β·β„Ž :(β„‚ Γ— β„‹)⟢ β„‹ β†’ Β·β„Ž Fn (β„‚ Γ— β„‹))
39 fnresdm 6670 . . . . . . . . . . . 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 4879 . . . . . . . . . 10 ⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩ = ⟨ +β„Ž , Β·β„Ž ⟩
42 normf 30407 . . . . . . . . . . 11 normβ„Ž: β„‹βŸΆβ„
43 ffn 6718 . . . . . . . . . . 11 (normβ„Ž: β„‹βŸΆβ„ β†’ normβ„Ž Fn β„‹)
44 fnresdm 6670 . . . . . . . . . . 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 4879 . . . . . . . . 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 2764 . . . . . . . 8 ⟨⟨( +β„Ž β†Ύ ( β„‹ Γ— β„‹)), ( Β·β„Ž β†Ύ (β„‚ Γ— β„‹))⟩, (normβ„Ž β†Ύ β„‹)⟩ = π‘ˆ
481hhnv 30449 . . . . . . . . 9 π‘ˆ ∈ NrmCVec
49 eqid 2733 . . . . . . . . . 10 (SubSpβ€˜π‘ˆ) = (SubSpβ€˜π‘ˆ)
5049sspid 30009 . . . . . . . . 9 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ ∈ (SubSpβ€˜π‘ˆ))
5148, 50ax-mp 5 . . . . . . . 8 π‘ˆ ∈ (SubSpβ€˜π‘ˆ)
5247, 51eqeltri 2830 . . . . . . 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 4005 . . . . . . 7 β„‹ βŠ† β„‹
5452, 53pm3.2i 472 . . . . . 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 4594 . . . . 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 485 . . . 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 487 . . . 4 if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) βŠ† β„‹
581, 7, 56, 57hhshsslem2 30552 . . 3 if((π‘Š ∈ (SubSpβ€˜π‘ˆ) ∧ 𝐻 βŠ† β„‹), 𝐻, β„‹) ∈ Sβ„‹
596, 58dedth 4587 . 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 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949  ifcif 4529  βŸ¨cop 4635   Γ— cxp 5675   β†Ύ cres 5679   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  β„‚cc 11108  β„cr 11109  NrmCVeccnv 29868  SubSpcss 30005   β„‹chba 30203   +β„Ž cva 30204   Β·β„Ž csm 30205  normβ„Žcno 30207   Sβ„‹ csh 30212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190  ax-hilex 30283  ax-hfvadd 30284  ax-hvcom 30285  ax-hvass 30286  ax-hv0cl 30287  ax-hvaddid 30288  ax-hfvmul 30289  ax-hvmulid 30290  ax-hvmulass 30291  ax-hvdistr1 30292  ax-hvdistr2 30293  ax-hvmul0 30294  ax-hfi 30363  ax-his1 30366  ax-his2 30367  ax-his3 30368  ax-his4 30369
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-icc 13331  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-lm 22733  df-haus 22819  df-grpo 29777  df-gid 29778  df-ginv 29779  df-gdiv 29780  df-ablo 29829  df-vc 29843  df-nv 29876  df-va 29879  df-ba 29880  df-sm 29881  df-0v 29882  df-vs 29883  df-nmcv 29884  df-ims 29885  df-ssp 30006  df-hnorm 30252  df-hba 30253  df-hvsub 30255  df-hlim 30256  df-sh 30491  df-ch 30505  df-ch0 30537
This theorem is referenced by:  hhsssh2  30554
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