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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ccfldextdgrr Structured version   Visualization version   GIF version
Theorem ccfldextdgrr 33035
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 33015 . . 3 β„‚fld/FldExtℝfld
2 extdgval 33021 . . 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 21378 . . . 4 ℝ = (Baseβ€˜β„fld)
54fveq2i 6893 . . 3 ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld))
65fveq2i 6893 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (dimβ€˜((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld)))
7 ccfldsrarelvec 33034 . . . 4 ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec
8 df-pr 4630 . . . . . 6 {1, i} = ({1} βˆͺ {i})
9 eqid 2730 . . . . . . . 8 (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
10 eqidd 2731 . . . . . . . . . 10 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜β„))
11 cnfld0 21169 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
1211a1i 11 . . . . . . . . . 10 (⊀ β†’ 0 = (0gβ€˜β„‚fld))
13 ax-resscn 11169 . . . . . . . . . . . 12 ℝ βŠ† β„‚
14 cnfldbas 21148 . . . . . . . . . . . 12 β„‚ = (Baseβ€˜β„‚fld)
1513, 14sseqtri 4017 . . . . . . . . . . 11 ℝ βŠ† (Baseβ€˜β„‚fld)
1615a1i 11 . . . . . . . . . 10 (⊀ β†’ ℝ βŠ† (Baseβ€˜β„‚fld))
1710, 12, 16sralmod0 20955 . . . . . . . . 9 (⊀ β†’ 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
1817mptru 1546 . . . . . . . 8 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
197a1i 11 . . . . . . . 8 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec)
20 ax-1cn 11170 . . . . . . . . . 10 1 ∈ β„‚
21 ax-1ne0 11181 . . . . . . . . . 10 1 β‰  0
2210, 16srabase 20937 . . . . . . . . . . . . 13 (⊀ β†’ (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
2322mptru 1546 . . . . . . . . . . . 12 (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2414, 23eqtri 2758 . . . . . . . . . . 11 β„‚ = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2524, 18lindssn 32768 . . . . . . . . . 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 1459 . . . . . . . . 9 {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2726a1i 11 . . . . . . . 8 (⊀ β†’ {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
28 ax-icn 11171 . . . . . . . . . 10 i ∈ β„‚
29 ine0 11653 . . . . . . . . . 10 i β‰  0
3024, 18lindssn 32768 . . . . . . . . . 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 1459 . . . . . . . . 9 {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3231a1i 11 . . . . . . . 8 (⊀ β†’ {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
33 lveclmod 20861 . . . . . . . . . . . . . . 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 21377 . . . . . . . . . . . . . . . 16 ℝfld = (β„‚fld β†Ύs ℝ)
3610, 16srasca 20943 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
3736mptru 1546 . . . . . . . . . . . . . . . 16 (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3835, 37eqtri 2758 . . . . . . . . . . . . . . 15 ℝfld = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
39 cnfldmul 21150 . . . . . . . . . . . . . . . 16 Β· = (.rβ€˜β„‚fld)
4010, 16sravsca 20945 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„)))
4140mptru 1546 . . . . . . . . . . . . . . . 16 (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4239, 41eqtri 2758 . . . . . . . . . . . . . . 15 Β· = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4338, 4, 24, 42, 9lspsnel 20758 . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . 13 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ↔ βˆƒπ‘₯ ∈ ℝ 𝑧 = (π‘₯ Β· 1))
4538, 4, 24, 42, 9lspsnel 20758 . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . 13 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i}) ↔ βˆƒπ‘¦ ∈ ℝ 𝑧 = (𝑦 Β· i))
4744, 46anbi12i 625 . . . . . . . . . . . 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 3224 . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (π‘₯ Β· 1))
50 simpll 763 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ π‘₯ ∈ ℝ)
5150recnd 11246 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ π‘₯ ∈ β„‚)
5251mulridd 11235 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (π‘₯ Β· 1) = π‘₯)
5349, 52eqtrd 2770 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = π‘₯)
5453negeqd 11458 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -𝑧 = -π‘₯)
55 simprr 769 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (𝑦 Β· i))
56 simplr 765 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑦 ∈ ℝ)
5756recnd 11246 . . . . . . . . . . . . . . . . . . . . . . 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 11239 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑦 Β· i) = (i Β· 𝑦))
6055, 59eqtrd 2770 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (i Β· 𝑦))
6154, 60oveq12d 7429 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = (-π‘₯ + (i Β· 𝑦)))
6253, 51eqeltrd 2831 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 ∈ β„‚)
6362subidd 11563 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑧 βˆ’ 𝑧) = 0)
6463negeqd 11458 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = -0)
6562, 62negsubdid 11590 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = (-𝑧 + 𝑧))
66 neg0 11510 . . . . . . . . . . . . . . . . . . . . . 22 -0 = 0
6766a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = 0)
6864, 65, 673eqtr3d 2778 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = 0)
6961, 68eqtr3d 2772 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-π‘₯ + (i Β· 𝑦)) = 0)
7050renegcld 11645 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -π‘₯ ∈ ℝ)
71 creq0 32227 . . . . . . . . . . . . . . . . . . . 20 ((-π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((-π‘₯ = 0 ∧ 𝑦 = 0) ↔ (-π‘₯ + (i Β· 𝑦)) = 0))
7270, 56, 71syl2anc 582 . . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-π‘₯ = 0 ∧ 𝑦 = 0))
7473simpld 493 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -π‘₯ = 0)
7551, 74negcon1ad 11570 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = π‘₯)
7653, 75, 673eqtr2d 2776 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = 0)
7776ex 411 . . . . . . . . . . . . . 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 11221 . . . . . . . . . . . . . 14 (𝑧 = 0 β†’ 0 ∈ ℝ)
80 simpr 483 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ π‘₯ = 0)
8180oveq1d 7426 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (π‘₯ Β· 1) = (0 Β· 1))
8281eqeq2d 2741 . . . . . . . . . . . . . . . 16 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (𝑧 = (π‘₯ Β· 1) ↔ 𝑧 = (0 Β· 1)))
8382anbi1d 628 . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ 𝑦 = 0)
8685oveq1d 7426 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ (𝑦 Β· i) = (0 Β· i))
8786eqeq2d 2741 . . . . . . . . . . . . . . . 16 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ (𝑧 = (𝑦 Β· i) ↔ 𝑧 = (0 Β· i)))
8887anbi2d 627 . . . . . . . . . . . . . . 15 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ ((𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ (𝑧 = (0 Β· 1) ∧ 𝑧 = (0 Β· i))))
8920mul02i 11407 . . . . . . . . . . . . . . . . . 18 (0 Β· 1) = 0
9089eqeq2i 2743 . . . . . . . . . . . . . . . . 17 (𝑧 = (0 Β· 1) ↔ 𝑧 = 0)
9190biimpri 227 . . . . . . . . . . . . . . . 16 (𝑧 = 0 β†’ 𝑧 = (0 Β· 1))
9228mul02i 11407 . . . . . . . . . . . . . . . . . 18 (0 Β· i) = 0
9392eqeq2i 2743 . . . . . . . . . . . . . . . . 17 (𝑧 = (0 Β· i) ↔ 𝑧 = 0)
9493biimpri 227 . . . . . . . . . . . . . . . 16 (𝑧 = 0 β†’ 𝑧 = (0 Β· i))
9591, 94jca 510 . . . . . . . . . . . . . . 15 (𝑧 = 0 β†’ (𝑧 = (0 Β· 1) ∧ 𝑧 = (0 Β· i)))
9679, 88, 95rspcedvd 3613 . . . . . . . . . . . . . 14 (𝑧 = 0 β†’ βˆƒπ‘¦ ∈ ℝ (𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i)))
9779, 84, 96rspcedvd 3613 . . . . . . . . . . . . 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 298 . . . . . . . . . . 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 3963 . . . . . . . . . . 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 4643 . . . . . . . . . . 11 (𝑧 ∈ {0} ↔ 𝑧 = 0)
10299, 100, 1013bitr4i 302 . . . . . . . . . 10 (𝑧 ∈ (((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ∩ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i})) ↔ 𝑧 ∈ {0})
103102eqriv 2727 . . . . . . . . 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 32998 . . . . . . 7 (⊀ β†’ ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
106105mptru 1546 . . . . . 6 ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
1078, 106eqeltri 2827 . . . . 5 {1, i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
108 cnfldadd 21149 . . . . . . . . . 10 + = (+gβ€˜β„‚fld)
10910, 16sraaddg 20939 . . . . . . . . . . 11 (⊀ β†’ (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
110109mptru 1546 . . . . . . . . . 10 (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
111108, 110eqtri 2758 . . . . . . . . 9 + = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
11234a1i 11 . . . . . . . . 9 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod)
113 1cnd 11213 . . . . . . . . 9 (⊀ β†’ 1 ∈ β„‚)
11428a1i 11 . . . . . . . . 9 (⊀ β†’ i ∈ β„‚)
11524, 111, 38, 4, 42, 9, 112, 113, 114lspprel 20849 . . . . . . . 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 1546 . . . . . . 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 481 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ π‘₯ ∈ ℝ)
118117recnd 11246 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ π‘₯ ∈ β„‚)
119 1cnd 11213 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 1 ∈ β„‚)
120118, 119mulcld 11238 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (π‘₯ Β· 1) ∈ β„‚)
121 simpr 483 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ ℝ)
122121recnd 11246 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ β„‚)
12328a1i 11 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ i ∈ β„‚)
124122, 123mulcld 11238 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (𝑦 Β· i) ∈ β„‚)
125120, 124addcld 11237 . . . . . . . . . 10 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) ∈ β„‚)
126 eleq1 2819 . . . . . . . . . 10 (𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) β†’ (𝑧 ∈ β„‚ ↔ ((π‘₯ Β· 1) + (𝑦 Β· i)) ∈ β„‚))
127125, 126syl5ibrcom 246 . . . . . . . . 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 15061 . . . . . . . . 9 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ ℝ)
130 simpr 483 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ π‘₯ = (β„œβ€˜π‘§))
131130oveq1d 7426 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ (π‘₯ Β· 1) = ((β„œβ€˜π‘§) Β· 1))
132131oveq1d 7426 . . . . . . . . . . 11 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)))
133132eqeq2d 2741 . . . . . . . . . 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 15062 . . . . . . . . . 10 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ ℝ)
136 simpr 483 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ 𝑦 = (β„‘β€˜π‘§))
137136oveq1d 7426 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ (𝑦 Β· i) = ((β„‘β€˜π‘§) Β· i))
138137oveq2d 7427 . . . . . . . . . . 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 2741 . . . . . . . . . 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 15067 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ 𝑧 = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
141129recnd 11246 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ β„‚)
142141mulridd 11235 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„œβ€˜π‘§) Β· 1) = (β„œβ€˜π‘§))
143135recnd 11246 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ β„‚)
14428a1i 11 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ i ∈ β„‚)
145143, 144mulcomd 11239 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„‘β€˜π‘§) Β· i) = (i Β· (β„‘β€˜π‘§)))
146142, 145oveq12d 7429 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)) = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
147140, 146eqtr4d 2773 . . . . . . . . . 10 (𝑧 ∈ β„‚ β†’ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)))
148135, 139, 147rspcedvd 3613 . . . . . . . . 9 (𝑧 ∈ β„‚ β†’ βˆƒπ‘¦ ∈ ℝ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)))
149129, 134, 148rspcedvd 3613 . . . . . . . 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 274 . . . . . 6 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) ↔ 𝑧 ∈ β„‚)
152151eqriv 2727 . . . . 5 ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) = β„‚
153 eqid 2730 . . . . . 6 (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
15424, 153, 9islbs4 21606 . . . . 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 707 . . . 4 {1, i} ∈ (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
156153dimval 32973 . . . 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 688 . . 3 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (β™―β€˜{1, i})
158 1nei 32228 . . . 4 1 β‰  i
159 hashprg 14359 . . . . 5 ((1 ∈ β„‚ ∧ i ∈ β„‚) β†’ (1 β‰  i ↔ (β™―β€˜{1, i}) = 2))
16020, 28, 159mp2an 688 . . . 4 (1 β‰  i ↔ (β™―β€˜{1, i}) = 2)
161158, 160mpbi 229 . . 3 (β™―β€˜{1, i}) = 2
162157, 161eqtri 2758 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = 2
1633, 6, 1623eqtr2i 2764 1 (β„‚fld[:]ℝfld) = 2
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539  βŠ€wtru 1540   ∈ wcel 2104   β‰  wne 2938  βˆƒwrex 3068   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  {csn 4627  {cpr 4629   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113  ici 11114   + caddc 11115   Β· cmul 11117   βˆ’ cmin 11448  -cneg 11449  2c2 12271  β™―chash 14294  β„œcre 15048  β„‘cim 15049  Basecbs 17148   β†Ύs cress 17177  +gcplusg 17201  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389  LModclmod 20614  LSpanclspn 20726  LBasisclbs 20829  LVecclvec 20857  subringAlg csra 20926  β„‚fldccnfld 21144  β„fldcrefld 21376  LIndSclinds 21579  dimcldim 32971  /FldExtcfldext 33005  [:]cextdg 33008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460  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-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-r1 9761  df-rank 9762  df-dju 9898  df-card 9936  df-acn 9939  df-ac 10113  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-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ocomp 17222  df-ds 17223  df-unif 17224  df-0g 17391  df-mre 17534  df-mrc 17535  df-mri 17536  df-acs 17537  df-proset 18252  df-drs 18253  df-poset 18270  df-ipo 18485  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-cntz 19222  df-lsm 19545  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-dvr 20292  df-subrng 20434  df-subrg 20459  df-drng 20502  df-field 20503  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lbs 20830  df-lvec 20858  df-sra 20930  df-cnfld 21145  df-refld 21377  df-lindf 21580  df-linds 21581  df-dim 32972  df-fldext 33009  df-extdg 33010
This theorem is referenced by: (None)
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