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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ccfldextdgrr Structured version   Visualization version   GIF version

Theorem ccfldextdgrr 32396
Description: The degree of the field extension of the complex numbers over the real numbers is 2. (Suggested by GL, 4-Aug-2023.) (Contributed by Thierry Arnoux, 20-Aug-2023.)
Assertion
Ref Expression
ccfldextdgrr (β„‚fld[:]ℝfld) = 2

Proof of Theorem ccfldextdgrr
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ccfldextrr 32377 . . 3 β„‚fld/FldExtℝfld
2 extdgval 32383 . . 3 (β„‚fld/FldExtℝfld β†’ (β„‚fld[:]ℝfld) = (dimβ€˜((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld))))
31, 2ax-mp 5 . 2 (β„‚fld[:]ℝfld) = (dimβ€˜((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld)))
4 rebase 21026 . . . 4 ℝ = (Baseβ€˜β„fld)
54fveq2i 6850 . . 3 ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld))
65fveq2i 6850 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (dimβ€˜((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld)))
7 ccfldsrarelvec 32395 . . . 4 ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec
8 df-pr 4594 . . . . . 6 {1, i} = ({1} βˆͺ {i})
9 eqid 2737 . . . . . . . 8 (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
10 eqidd 2738 . . . . . . . . . 10 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜β„))
11 cnfld0 20837 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
1211a1i 11 . . . . . . . . . 10 (⊀ β†’ 0 = (0gβ€˜β„‚fld))
13 ax-resscn 11115 . . . . . . . . . . . 12 ℝ βŠ† β„‚
14 cnfldbas 20816 . . . . . . . . . . . 12 β„‚ = (Baseβ€˜β„‚fld)
1513, 14sseqtri 3985 . . . . . . . . . . 11 ℝ βŠ† (Baseβ€˜β„‚fld)
1615a1i 11 . . . . . . . . . 10 (⊀ β†’ ℝ βŠ† (Baseβ€˜β„‚fld))
1710, 12, 16sralmod0 20673 . . . . . . . . 9 (⊀ β†’ 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
1817mptru 1549 . . . . . . . 8 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
197a1i 11 . . . . . . . 8 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec)
20 ax-1cn 11116 . . . . . . . . . 10 1 ∈ β„‚
21 ax-1ne0 11127 . . . . . . . . . 10 1 β‰  0
2210, 16srabase 20656 . . . . . . . . . . . . 13 (⊀ β†’ (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
2322mptru 1549 . . . . . . . . . . . 12 (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2414, 23eqtri 2765 . . . . . . . . . . 11 β„‚ = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2524, 18lindssn 32206 . . . . . . . . . 10 ((((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec ∧ 1 ∈ β„‚ ∧ 1 β‰  0) β†’ {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
267, 20, 21, 25mp3an 1462 . . . . . . . . 9 {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2726a1i 11 . . . . . . . 8 (⊀ β†’ {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
28 ax-icn 11117 . . . . . . . . . 10 i ∈ β„‚
29 ine0 11597 . . . . . . . . . 10 i β‰  0
3024, 18lindssn 32206 . . . . . . . . . 10 ((((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec ∧ i ∈ β„‚ ∧ i β‰  0) β†’ {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
317, 28, 29, 30mp3an 1462 . . . . . . . . 9 {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3231a1i 11 . . . . . . . 8 (⊀ β†’ {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
33 lveclmod 20583 . . . . . . . . . . . . . . 15 (((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod)
347, 33ax-mp 5 . . . . . . . . . . . . . 14 ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod
35 df-refld 21025 . . . . . . . . . . . . . . . 16 ℝfld = (β„‚fld β†Ύs ℝ)
3610, 16srasca 20662 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
3736mptru 1549 . . . . . . . . . . . . . . . 16 (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3835, 37eqtri 2765 . . . . . . . . . . . . . . 15 ℝfld = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
39 cnfldmul 20818 . . . . . . . . . . . . . . . 16 Β· = (.rβ€˜β„‚fld)
4010, 16sravsca 20664 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„)))
4140mptru 1549 . . . . . . . . . . . . . . . 16 (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4239, 41eqtri 2765 . . . . . . . . . . . . . . 15 Β· = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4338, 4, 24, 42, 9lspsnel 20480 . . . . . . . . . . . . . 14 ((((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod ∧ 1 ∈ β„‚) β†’ (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ↔ βˆƒπ‘₯ ∈ ℝ 𝑧 = (π‘₯ Β· 1)))
4434, 20, 43mp2an 691 . . . . . . . . . . . . 13 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ↔ βˆƒπ‘₯ ∈ ℝ 𝑧 = (π‘₯ Β· 1))
4538, 4, 24, 42, 9lspsnel 20480 . . . . . . . . . . . . . 14 ((((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod ∧ i ∈ β„‚) β†’ (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i}) ↔ βˆƒπ‘¦ ∈ ℝ 𝑧 = (𝑦 Β· i)))
4634, 28, 45mp2an 691 . . . . . . . . . . . . 13 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i}) ↔ βˆƒπ‘¦ ∈ ℝ 𝑧 = (𝑦 Β· i))
4744, 46anbi12i 628 . . . . . . . . . . . 12 ((𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∧ 𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})) ↔ (βˆƒπ‘₯ ∈ ℝ 𝑧 = (π‘₯ Β· 1) ∧ βˆƒπ‘¦ ∈ ℝ 𝑧 = (𝑦 Β· i)))
48 reeanv 3220 . . . . . . . . . . . 12 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ (βˆƒπ‘₯ ∈ ℝ 𝑧 = (π‘₯ Β· 1) ∧ βˆƒπ‘¦ ∈ ℝ 𝑧 = (𝑦 Β· i)))
49 simprl 770 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (π‘₯ Β· 1))
50 simpll 766 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ π‘₯ ∈ ℝ)
5150recnd 11190 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ π‘₯ ∈ β„‚)
5251mulid1d 11179 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (π‘₯ Β· 1) = π‘₯)
5349, 52eqtrd 2777 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = π‘₯)
5453negeqd 11402 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -𝑧 = -π‘₯)
55 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (𝑦 Β· i))
56 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑦 ∈ ℝ)
5756recnd 11190 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑦 ∈ β„‚)
5828a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ i ∈ β„‚)
5957, 58mulcomd 11183 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑦 Β· i) = (i Β· 𝑦))
6055, 59eqtrd 2777 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (i Β· 𝑦))
6154, 60oveq12d 7380 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = (-π‘₯ + (i Β· 𝑦)))
6253, 51eqeltrd 2838 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 ∈ β„‚)
6362subidd 11507 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑧 βˆ’ 𝑧) = 0)
6463negeqd 11402 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = -0)
6562, 62negsubdid 11534 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = (-𝑧 + 𝑧))
66 neg0 11454 . . . . . . . . . . . . . . . . . . . . . 22 -0 = 0
6766a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = 0)
6864, 65, 673eqtr3d 2785 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = 0)
6961, 68eqtr3d 2779 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-π‘₯ + (i Β· 𝑦)) = 0)
7050renegcld 11589 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -π‘₯ ∈ ℝ)
71 creq0 31694 . . . . . . . . . . . . . . . . . . . 20 ((-π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((-π‘₯ = 0 ∧ 𝑦 = 0) ↔ (-π‘₯ + (i Β· 𝑦)) = 0))
7270, 56, 71syl2anc 585 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ ((-π‘₯ = 0 ∧ 𝑦 = 0) ↔ (-π‘₯ + (i Β· 𝑦)) = 0))
7369, 72mpbird 257 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-π‘₯ = 0 ∧ 𝑦 = 0))
7473simpld 496 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -π‘₯ = 0)
7551, 74negcon1ad 11514 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = π‘₯)
7653, 75, 673eqtr2d 2783 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = 0)
7776ex 414 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) β†’ 𝑧 = 0))
7877rexlimivv 3197 . . . . . . . . . . . . 13 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) β†’ 𝑧 = 0)
79 0red 11165 . . . . . . . . . . . . . 14 (𝑧 = 0 β†’ 0 ∈ ℝ)
80 simpr 486 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ π‘₯ = 0)
8180oveq1d 7377 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (π‘₯ Β· 1) = (0 Β· 1))
8281eqeq2d 2748 . . . . . . . . . . . . . . . 16 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (𝑧 = (π‘₯ Β· 1) ↔ 𝑧 = (0 Β· 1)))
8382anbi1d 631 . . . . . . . . . . . . . . 15 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ ((𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ (𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i))))
8483rexbidv 3176 . . . . . . . . . . . . . 14 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (βˆƒπ‘¦ ∈ ℝ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ βˆƒπ‘¦ ∈ ℝ (𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i))))
85 simpr 486 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ 𝑦 = 0)
8685oveq1d 7377 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ (𝑦 Β· i) = (0 Β· i))
8786eqeq2d 2748 . . . . . . . . . . . . . . . 16 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ (𝑧 = (𝑦 Β· i) ↔ 𝑧 = (0 Β· i)))
8887anbi2d 630 . . . . . . . . . . . . . . 15 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ ((𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ (𝑧 = (0 Β· 1) ∧ 𝑧 = (0 Β· i))))
8920mul02i 11351 . . . . . . . . . . . . . . . . . 18 (0 Β· 1) = 0
9089eqeq2i 2750 . . . . . . . . . . . . . . . . 17 (𝑧 = (0 Β· 1) ↔ 𝑧 = 0)
9190biimpri 227 . . . . . . . . . . . . . . . 16 (𝑧 = 0 β†’ 𝑧 = (0 Β· 1))
9228mul02i 11351 . . . . . . . . . . . . . . . . . 18 (0 Β· i) = 0
9392eqeq2i 2750 . . . . . . . . . . . . . . . . 17 (𝑧 = (0 Β· i) ↔ 𝑧 = 0)
9493biimpri 227 . . . . . . . . . . . . . . . 16 (𝑧 = 0 β†’ 𝑧 = (0 Β· i))
9591, 94jca 513 . . . . . . . . . . . . . . 15 (𝑧 = 0 β†’ (𝑧 = (0 Β· 1) ∧ 𝑧 = (0 Β· i)))
9679, 88, 95rspcedvd 3586 . . . . . . . . . . . . . 14 (𝑧 = 0 β†’ βˆƒπ‘¦ ∈ ℝ (𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i)))
9779, 84, 96rspcedvd 3586 . . . . . . . . . . . . 13 (𝑧 = 0 β†’ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)))
9878, 97impbii 208 . . . . . . . . . . . 12 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ 𝑧 = 0)
9947, 48, 983bitr2i 299 . . . . . . . . . . 11 ((𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∧ 𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})) ↔ 𝑧 = 0)
100 elin 3931 . . . . . . . . . . 11 (𝑧 ∈ (((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∩ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})) ↔ (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∧ 𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})))
101 velsn 4607 . . . . . . . . . . 11 (𝑧 ∈ {0} ↔ 𝑧 = 0)
10299, 100, 1013bitr4i 303 . . . . . . . . . 10 (𝑧 ∈ (((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∩ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})) ↔ 𝑧 ∈ {0})
103102eqriv 2734 . . . . . . . . 9 (((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∩ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})) = {0}
104103a1i 11 . . . . . . . 8 (⊀ β†’ (((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∩ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})) = {0})
1059, 18, 19, 27, 32, 104lindsun 32360 . . . . . . 7 (⊀ β†’ ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
106105mptru 1549 . . . . . 6 ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
1078, 106eqeltri 2834 . . . . 5 {1, i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
108 cnfldadd 20817 . . . . . . . . . 10 + = (+gβ€˜β„‚fld)
10910, 16sraaddg 20658 . . . . . . . . . . 11 (⊀ β†’ (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
110109mptru 1549 . . . . . . . . . 10 (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
111108, 110eqtri 2765 . . . . . . . . 9 + = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
11234a1i 11 . . . . . . . . 9 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod)
113 1cnd 11157 . . . . . . . . 9 (⊀ β†’ 1 ∈ β„‚)
11428a1i 11 . . . . . . . . 9 (⊀ β†’ i ∈ β„‚)
11524, 111, 38, 4, 42, 9, 112, 113, 114lspprel 20571 . . . . . . . 8 (⊀ β†’ (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i))))
116115mptru 1549 . . . . . . 7 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) ↔ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)))
117 simpl 484 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ π‘₯ ∈ ℝ)
118117recnd 11190 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ π‘₯ ∈ β„‚)
119 1cnd 11157 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 1 ∈ β„‚)
120118, 119mulcld 11182 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (π‘₯ Β· 1) ∈ β„‚)
121 simpr 486 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ ℝ)
122121recnd 11190 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ β„‚)
12328a1i 11 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ i ∈ β„‚)
124122, 123mulcld 11182 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (𝑦 Β· i) ∈ β„‚)
125120, 124addcld 11181 . . . . . . . . . 10 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) ∈ β„‚)
126 eleq1 2826 . . . . . . . . . 10 (𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) β†’ (𝑧 ∈ β„‚ ↔ ((π‘₯ Β· 1) + (𝑦 Β· i)) ∈ β„‚))
127125, 126syl5ibrcom 247 . . . . . . . . 9 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) β†’ 𝑧 ∈ β„‚))
128127rexlimivv 3197 . . . . . . . 8 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) β†’ 𝑧 ∈ β„‚)
129 recl 15002 . . . . . . . . 9 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ ℝ)
130 simpr 486 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ π‘₯ = (β„œβ€˜π‘§))
131130oveq1d 7377 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ (π‘₯ Β· 1) = ((β„œβ€˜π‘§) Β· 1))
132131oveq1d 7377 . . . . . . . . . . 11 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)))
133132eqeq2d 2748 . . . . . . . . . 10 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ (𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) ↔ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i))))
134133rexbidv 3176 . . . . . . . . 9 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ (βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) ↔ βˆƒπ‘¦ ∈ ℝ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i))))
135 imcl 15003 . . . . . . . . . 10 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ ℝ)
136 simpr 486 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ 𝑦 = (β„‘β€˜π‘§))
137136oveq1d 7377 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ (𝑦 Β· i) = ((β„‘β€˜π‘§) Β· i))
138137oveq2d 7378 . . . . . . . . . . 11 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)) = (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)))
139138eqeq2d 2748 . . . . . . . . . 10 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ (𝑧 = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)) ↔ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i))))
140 replim 15008 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ 𝑧 = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
141129recnd 11190 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ β„‚)
142141mulid1d 11179 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„œβ€˜π‘§) Β· 1) = (β„œβ€˜π‘§))
143135recnd 11190 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ β„‚)
14428a1i 11 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ i ∈ β„‚)
145143, 144mulcomd 11183 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„‘β€˜π‘§) Β· i) = (i Β· (β„‘β€˜π‘§)))
146142, 145oveq12d 7380 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)) = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
147140, 146eqtr4d 2780 . . . . . . . . . 10 (𝑧 ∈ β„‚ β†’ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)))
148135, 139, 147rspcedvd 3586 . . . . . . . . 9 (𝑧 ∈ β„‚ β†’ βˆƒπ‘¦ ∈ ℝ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)))
149129, 134, 148rspcedvd 3586 . . . . . . . 8 (𝑧 ∈ β„‚ β†’ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)))
150128, 149impbii 208 . . . . . . 7 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) ↔ 𝑧 ∈ β„‚)
151116, 150bitri 275 . . . . . 6 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) ↔ 𝑧 ∈ β„‚)
152151eqriv 2734 . . . . 5 ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) = β„‚
153 eqid 2737 . . . . . 6 (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
15424, 153, 9islbs4 21254 . . . . 5 ({1, i} ∈ (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) ↔ ({1, i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) ∧ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) = β„‚))
155107, 152, 154mpbir2an 710 . . . 4 {1, i} ∈ (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
156153dimval 32340 . . . 4 ((((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec ∧ {1, i} ∈ (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))) β†’ (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (β™―β€˜{1, i}))
1577, 155, 156mp2an 691 . . 3 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (β™―β€˜{1, i})
158 1nei 31695 . . . 4 1 β‰  i
159 hashprg 14302 . . . . 5 ((1 ∈ β„‚ ∧ i ∈ β„‚) β†’ (1 β‰  i ↔ (β™―β€˜{1, i}) = 2))
16020, 28, 159mp2an 691 . . . 4 (1 β‰  i ↔ (β™―β€˜{1, i}) = 2)
161158, 160mpbi 229 . . 3 (β™―β€˜{1, i}) = 2
162157, 161eqtri 2765 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = 2
1633, 6, 1623eqtr2i 2771 1 (β„‚fld[:]ℝfld) = 2
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  βŠ€wtru 1543   ∈ wcel 2107   β‰  wne 2944  βˆƒwrex 3074   βˆͺ cun 3913   ∩ cin 3914   βŠ† wss 3915  {csn 4591  {cpr 4593   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059  ici 11060   + caddc 11061   Β· cmul 11063   βˆ’ cmin 11392  -cneg 11393  2c2 12215  β™―chash 14237  β„œcre 14989  β„‘cim 14990  Basecbs 17090   β†Ύs cress 17119  +gcplusg 17140  .rcmulr 17141  Scalarcsca 17143   ·𝑠 cvsca 17144  0gc0g 17328  LModclmod 20338  LSpanclspn 20448  LBasisclbs 20551  LVecclvec 20579  subringAlg csra 20645  β„‚fldccnfld 20812  β„fldcrefld 21024  LIndSclinds 21227  dimcldim 32338  /FldExtcfldext 32367  [:]cextdg 32370
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-reg 9535  ax-inf2 9584  ax-ac2 10406  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-addf 11137  ax-mulf 11138
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9453  df-r1 9707  df-rank 9708  df-dju 9844  df-card 9882  df-acn 9885  df-ac 10059  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-xnn0 12493  df-z 12507  df-dec 12626  df-uz 12771  df-fz 13432  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-mulr 17154  df-starv 17155  df-sca 17156  df-vsca 17157  df-ip 17158  df-tset 17159  df-ple 17160  df-ocomp 17161  df-ds 17162  df-unif 17163  df-0g 17330  df-mre 17473  df-mrc 17474  df-mri 17475  df-acs 17476  df-proset 18191  df-drs 18192  df-poset 18209  df-ipo 18424  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-grp 18758  df-minusg 18759  df-sbg 18760  df-subg 18932  df-cntz 19104  df-lsm 19425  df-cmn 19571  df-abl 19572  df-mgp 19904  df-ur 19921  df-ring 19973  df-cring 19974  df-oppr 20056  df-dvdsr 20077  df-unit 20078  df-invr 20108  df-dvr 20119  df-drng 20201  df-field 20202  df-subrg 20236  df-lmod 20340  df-lss 20409  df-lsp 20449  df-lbs 20552  df-lvec 20580  df-sra 20649  df-cnfld 20813  df-refld 21025  df-lindf 21228  df-linds 21229  df-dim 32339  df-fldext 32371  df-extdg 32372
This theorem is referenced by: (None)
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