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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ccfldextdgrr Structured version   Visualization version   GIF version
Theorem ccfldextdgrr 33430
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 33410 . . 3 β„‚fld/FldExtℝfld
2 extdgval 33416 . . 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 21542 . . . 4 ℝ = (Baseβ€˜β„fld)
54fveq2i 6897 . . 3 ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld))
65fveq2i 6897 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (dimβ€˜((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld)))
7 ccfldsrarelvec 33429 . . . 4 ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec
8 df-pr 4632 . . . . . 6 {1, i} = ({1} βˆͺ {i})
9 eqid 2725 . . . . . . . 8 (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
10 eqidd 2726 . . . . . . . . . 10 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜β„))
11 cnfld0 21324 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
1211a1i 11 . . . . . . . . . 10 (⊀ β†’ 0 = (0gβ€˜β„‚fld))
13 ax-resscn 11195 . . . . . . . . . . . 12 ℝ βŠ† β„‚
14 cnfldbas 21287 . . . . . . . . . . . 12 β„‚ = (Baseβ€˜β„‚fld)
1513, 14sseqtri 4014 . . . . . . . . . . 11 ℝ βŠ† (Baseβ€˜β„‚fld)
1615a1i 11 . . . . . . . . . 10 (⊀ β†’ ℝ βŠ† (Baseβ€˜β„‚fld))
1710, 12, 16sralmod0 21085 . . . . . . . . 9 (⊀ β†’ 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
1817mptru 1540 . . . . . . . 8 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
197a1i 11 . . . . . . . 8 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec)
20 ax-1cn 11196 . . . . . . . . . 10 1 ∈ β„‚
21 ax-1ne0 11207 . . . . . . . . . 10 1 β‰  0
2210, 16srabase 21067 . . . . . . . . . . . . 13 (⊀ β†’ (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
2322mptru 1540 . . . . . . . . . . . 12 (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2414, 23eqtri 2753 . . . . . . . . . . 11 β„‚ = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2524, 18lindssn 33155 . . . . . . . . . 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 1457 . . . . . . . . 9 {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2726a1i 11 . . . . . . . 8 (⊀ β†’ {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
28 ax-icn 11197 . . . . . . . . . 10 i ∈ β„‚
29 ine0 11679 . . . . . . . . . 10 i β‰  0
3024, 18lindssn 33155 . . . . . . . . . 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 1457 . . . . . . . . 9 {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3231a1i 11 . . . . . . . 8 (⊀ β†’ {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
33 lveclmod 20995 . . . . . . . . . . . . . . 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 21541 . . . . . . . . . . . . . . . 16 ℝfld = (β„‚fld β†Ύs ℝ)
3610, 16srasca 21073 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
3736mptru 1540 . . . . . . . . . . . . . . . 16 (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3835, 37eqtri 2753 . . . . . . . . . . . . . . 15 ℝfld = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
39 cnfldmul 21291 . . . . . . . . . . . . . . . 16 Β· = (.rβ€˜β„‚fld)
4010, 16sravsca 21075 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„)))
4140mptru 1540 . . . . . . . . . . . . . . . 16 (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4239, 41eqtri 2753 . . . . . . . . . . . . . . 15 Β· = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4338, 4, 24, 42, 9lspsnel 20891 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . 13 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1}) ↔ βˆƒπ‘₯ ∈ ℝ 𝑧 = (π‘₯ Β· 1))
4538, 4, 24, 42, 9lspsnel 20891 . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . 13 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{i}) ↔ βˆƒπ‘¦ ∈ ℝ 𝑧 = (𝑦 Β· i))
4744, 46anbi12i 626 . . . . . . . . . . . 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 3217 . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (π‘₯ Β· 1))
50 simpll 765 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ π‘₯ ∈ ℝ)
5150recnd 11272 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ π‘₯ ∈ β„‚)
5251mulridd 11261 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (π‘₯ Β· 1) = π‘₯)
5349, 52eqtrd 2765 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = π‘₯)
5453negeqd 11484 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -𝑧 = -π‘₯)
55 simprr 771 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (𝑦 Β· i))
56 simplr 767 . . . . . . . . . . . . . . . . . . . . . . . 24 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑦 ∈ ℝ)
5756recnd 11272 . . . . . . . . . . . . . . . . . . . . . . 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 11265 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑦 Β· i) = (i Β· 𝑦))
6055, 59eqtrd 2765 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (i Β· 𝑦))
6154, 60oveq12d 7435 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = (-π‘₯ + (i Β· 𝑦)))
6253, 51eqeltrd 2825 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 ∈ β„‚)
6362subidd 11589 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑧 βˆ’ 𝑧) = 0)
6463negeqd 11484 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = -0)
6562, 62negsubdid 11616 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = (-𝑧 + 𝑧))
66 neg0 11536 . . . . . . . . . . . . . . . . . . . . . 22 -0 = 0
6766a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = 0)
6864, 65, 673eqtr3d 2773 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = 0)
6961, 68eqtr3d 2767 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-π‘₯ + (i Β· 𝑦)) = 0)
7050renegcld 11671 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -π‘₯ ∈ ℝ)
71 creq0 32574 . . . . . . . . . . . . . . . . . . . 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 11596 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = π‘₯)
7653, 75, 673eqtr2d 2771 . . . . . . . . . . . . . . 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 3190 . . . . . . . . . . . . 13 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) β†’ 𝑧 = 0)
79 0red 11247 . . . . . . . . . . . . . 14 (𝑧 = 0 β†’ 0 ∈ ℝ)
80 simpr 483 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ π‘₯ = 0)
8180oveq1d 7432 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (π‘₯ Β· 1) = (0 Β· 1))
8281eqeq2d 2736 . . . . . . . . . . . . . . . 16 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (𝑧 = (π‘₯ Β· 1) ↔ 𝑧 = (0 Β· 1)))
8382anbi1d 629 . . . . . . . . . . . . . . 15 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ ((𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ (𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i))))
8483rexbidv 3169 . . . . . . . . . . . . . 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 7432 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ (𝑦 Β· i) = (0 Β· i))
8786eqeq2d 2736 . . . . . . . . . . . . . . . 16 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ (𝑧 = (𝑦 Β· i) ↔ 𝑧 = (0 Β· i)))
8887anbi2d 628 . . . . . . . . . . . . . . 15 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ ((𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i)) ↔ (𝑧 = (0 Β· 1) ∧ 𝑧 = (0 Β· i))))
8920mul02i 11433 . . . . . . . . . . . . . . . . . 18 (0 Β· 1) = 0
9089eqeq2i 2738 . . . . . . . . . . . . . . . . 17 (𝑧 = (0 Β· 1) ↔ 𝑧 = 0)
9190biimpri 227 . . . . . . . . . . . . . . . 16 (𝑧 = 0 β†’ 𝑧 = (0 Β· 1))
9228mul02i 11433 . . . . . . . . . . . . . . . . . 18 (0 Β· i) = 0
9392eqeq2i 2738 . . . . . . . . . . . . . . . . 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 3609 . . . . . . . . . . . . . 14 (𝑧 = 0 β†’ βˆƒπ‘¦ ∈ ℝ (𝑧 = (0 Β· 1) ∧ 𝑧 = (𝑦 Β· i)))
9779, 84, 96rspcedvd 3609 . . . . . . . . . . . . 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 3961 . . . . . . . . . . 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 4645 . . . . . . . . . . 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 2722 . . . . . . . . 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 33393 . . . . . . 7 (⊀ β†’ ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
106105mptru 1540 . . . . . 6 ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
1078, 106eqeltri 2821 . . . . 5 {1, i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
108 cnfldadd 21289 . . . . . . . . . 10 + = (+gβ€˜β„‚fld)
10910, 16sraaddg 21069 . . . . . . . . . . 11 (⊀ β†’ (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
110109mptru 1540 . . . . . . . . . 10 (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
111108, 110eqtri 2753 . . . . . . . . 9 + = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
11234a1i 11 . . . . . . . . 9 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod)
113 1cnd 11239 . . . . . . . . 9 (⊀ β†’ 1 ∈ β„‚)
11428a1i 11 . . . . . . . . 9 (⊀ β†’ i ∈ β„‚)
11524, 111, 38, 4, 42, 9, 112, 113, 114lspprel 20983 . . . . . . . 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 1540 . . . . . . 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 11272 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ π‘₯ ∈ β„‚)
119 1cnd 11239 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 1 ∈ β„‚)
120118, 119mulcld 11264 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (π‘₯ Β· 1) ∈ β„‚)
121 simpr 483 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ ℝ)
122121recnd 11272 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ β„‚)
12328a1i 11 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ i ∈ β„‚)
124122, 123mulcld 11264 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (𝑦 Β· i) ∈ β„‚)
125120, 124addcld 11263 . . . . . . . . . 10 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) ∈ β„‚)
126 eleq1 2813 . . . . . . . . . 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 3190 . . . . . . . 8 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) β†’ 𝑧 ∈ β„‚)
129 recl 15089 . . . . . . . . 9 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ ℝ)
130 simpr 483 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ π‘₯ = (β„œβ€˜π‘§))
131130oveq1d 7432 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ (π‘₯ Β· 1) = ((β„œβ€˜π‘§) Β· 1))
132131oveq1d 7432 . . . . . . . . . . 11 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)))
133132eqeq2d 2736 . . . . . . . . . 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 3169 . . . . . . . . 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 15090 . . . . . . . . . 10 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ ℝ)
136 simpr 483 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ 𝑦 = (β„‘β€˜π‘§))
137136oveq1d 7432 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ (𝑦 Β· i) = ((β„‘β€˜π‘§) Β· i))
138137oveq2d 7433 . . . . . . . . . . 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 2736 . . . . . . . . . 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 15095 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ 𝑧 = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
141129recnd 11272 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ β„‚)
142141mulridd 11261 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„œβ€˜π‘§) Β· 1) = (β„œβ€˜π‘§))
143135recnd 11272 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ β„‚)
14428a1i 11 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ i ∈ β„‚)
145143, 144mulcomd 11265 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„‘β€˜π‘§) Β· i) = (i Β· (β„‘β€˜π‘§)))
146142, 145oveq12d 7435 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)) = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
147140, 146eqtr4d 2768 . . . . . . . . . 10 (𝑧 ∈ β„‚ β†’ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)))
148135, 139, 147rspcedvd 3609 . . . . . . . . 9 (𝑧 ∈ β„‚ β†’ βˆƒπ‘¦ ∈ ℝ 𝑧 = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)))
149129, 134, 148rspcedvd 3609 . . . . . . . 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 2722 . . . . 5 ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) = β„‚
153 eqid 2725 . . . . . 6 (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
15424, 153, 9islbs4 21770 . . . . 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 709 . . . 4 {1, i} ∈ (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
156153dimval 33368 . . . 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 690 . . 3 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (β™―β€˜{1, i})
158 1nei 32575 . . . 4 1 β‰  i
159 hashprg 14386 . . . . 5 ((1 ∈ β„‚ ∧ i ∈ β„‚) β†’ (1 β‰  i ↔ (β™―β€˜{1, i}) = 2))
16020, 28, 159mp2an 690 . . . 4 (1 β‰  i ↔ (β™―β€˜{1, i}) = 2)
161158, 160mpbi 229 . . 3 (β™―β€˜{1, i}) = 2
162157, 161eqtri 2753 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = 2
1633, 6, 1623eqtr2i 2759 1 (β„‚fld[:]ℝfld) = 2
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  βŠ€wtru 1534   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060   βˆͺ cun 3943   ∩ cin 3944   βŠ† wss 3945  {csn 4629  {cpr 4631   class class class wbr 5148  β€˜cfv 6547  (class class class)co 7417  β„‚cc 11136  β„cr 11137  0cc0 11138  1c1 11139  ici 11140   + caddc 11141   Β· cmul 11143   βˆ’ cmin 11474  -cneg 11475  2c2 12297  β™―chash 14321  β„œcre 15076  β„‘cim 15077  Basecbs 17179   β†Ύs cress 17208  +gcplusg 17232  .rcmulr 17233  Scalarcsca 17235   ·𝑠 cvsca 17236  0gc0g 17420  LModclmod 20747  LSpanclspn 20859  LBasisclbs 20963  LVecclvec 20991  subringAlg csra 21060  β„‚fldccnfld 21283  β„fldcrefld 21540  LIndSclinds 21743  dimcldim 33366  /FldExtcfldext 33400  [:]cextdg 33403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-reg 9615  ax-inf2 9664  ax-ac2 10486  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-addf 11217  ax-mulf 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  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 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-isom 6556  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-r1 9787  df-rank 9788  df-dju 9924  df-card 9962  df-acn 9965  df-ac 10139  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-xnn0 12575  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-starv 17247  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ocomp 17253  df-ds 17254  df-unif 17255  df-0g 17422  df-mre 17565  df-mrc 17566  df-mri 17567  df-acs 17568  df-proset 18286  df-drs 18287  df-poset 18304  df-ipo 18519  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18740  df-grp 18897  df-minusg 18898  df-sbg 18899  df-subg 19082  df-cntz 19272  df-lsm 19595  df-cmn 19741  df-abl 19742  df-mgp 20079  df-rng 20097  df-ur 20126  df-ring 20179  df-cring 20180  df-oppr 20277  df-dvdsr 20300  df-unit 20301  df-invr 20331  df-dvr 20344  df-subrng 20487  df-subrg 20512  df-drng 20630  df-field 20631  df-lmod 20749  df-lss 20820  df-lsp 20860  df-lbs 20964  df-lvec 20992  df-sra 21062  df-cnfld 21284  df-refld 21541  df-lindf 21744  df-linds 21745  df-dim 33367  df-fldext 33404  df-extdg 33405
This theorem is referenced by: (None)
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