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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ccfldextdgrr Structured version   Visualization version   GIF version
Theorem ccfldextdgrr 33292
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 33272 . . 3 β„‚fld/FldExtℝfld
2 extdgval 33278 . . 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 21525 . . . 4 ℝ = (Baseβ€˜β„fld)
54fveq2i 6894 . . 3 ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld))
65fveq2i 6894 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (dimβ€˜((subringAlg β€˜β„‚fld)β€˜(Baseβ€˜β„fld)))
7 ccfldsrarelvec 33291 . . . 4 ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec
8 df-pr 4627 . . . . . 6 {1, i} = ({1} βˆͺ {i})
9 eqid 2727 . . . . . . . 8 (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
10 eqidd 2728 . . . . . . . . . 10 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) = ((subringAlg β€˜β„‚fld)β€˜β„))
11 cnfld0 21307 . . . . . . . . . . 11 0 = (0gβ€˜β„‚fld)
1211a1i 11 . . . . . . . . . 10 (⊀ β†’ 0 = (0gβ€˜β„‚fld))
13 ax-resscn 11187 . . . . . . . . . . . 12 ℝ βŠ† β„‚
14 cnfldbas 21270 . . . . . . . . . . . 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 21070 . . . . . . . . 9 (⊀ β†’ 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
1817mptru 1541 . . . . . . . 8 0 = (0gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
197a1i 11 . . . . . . . 8 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LVec)
20 ax-1cn 11188 . . . . . . . . . 10 1 ∈ β„‚
21 ax-1ne0 11199 . . . . . . . . . 10 1 β‰  0
2210, 16srabase 21052 . . . . . . . . . . . . 13 (⊀ β†’ (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
2322mptru 1541 . . . . . . . . . . . 12 (Baseβ€˜β„‚fld) = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2414, 23eqtri 2755 . . . . . . . . . . 11 β„‚ = (Baseβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2524, 18lindssn 33033 . . . . . . . . . 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 1458 . . . . . . . . 9 {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
2726a1i 11 . . . . . . . 8 (⊀ β†’ {1} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
28 ax-icn 11189 . . . . . . . . . 10 i ∈ β„‚
29 ine0 11671 . . . . . . . . . 10 i β‰  0
3024, 18lindssn 33033 . . . . . . . . . 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 1458 . . . . . . . . 9 {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3231a1i 11 . . . . . . . 8 (⊀ β†’ {i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
33 lveclmod 20980 . . . . . . . . . . . . . . 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 21524 . . . . . . . . . . . . . . . 16 ℝfld = (β„‚fld β†Ύs ℝ)
3610, 16srasca 21058 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
3736mptru 1541 . . . . . . . . . . . . . . . 16 (β„‚fld β†Ύs ℝ) = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
3835, 37eqtri 2755 . . . . . . . . . . . . . . 15 ℝfld = (Scalarβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
39 cnfldmul 21274 . . . . . . . . . . . . . . . 16 Β· = (.rβ€˜β„‚fld)
4010, 16sravsca 21060 . . . . . . . . . . . . . . . . 17 (⊀ β†’ (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„)))
4140mptru 1541 . . . . . . . . . . . . . . . 16 (.rβ€˜β„‚fld) = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4239, 41eqtri 2755 . . . . . . . . . . . . . . 15 Β· = ( ·𝑠 β€˜((subringAlg β€˜β„‚fld)β€˜β„))
4338, 4, 24, 42, 9lspsnel 20876 . . . . . . . . . . . . . 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 20876 . . . . . . . . . . . . . 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 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 3221 . . . . . . . . . . . 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 11264 . . . . . . . . . . . . . . . . . 18 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ π‘₯ ∈ β„‚)
5251mulridd 11253 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (π‘₯ Β· 1) = π‘₯)
5349, 52eqtrd 2767 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = π‘₯)
5453negeqd 11476 . . . . . . . . . . . . . . . . . . . . 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 11264 . . . . . . . . . . . . . . . . . . . . . . 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 11257 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑦 Β· i) = (i Β· 𝑦))
6055, 59eqtrd 2767 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = (i Β· 𝑦))
6154, 60oveq12d 7432 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = (-π‘₯ + (i Β· 𝑦)))
6253, 51eqeltrd 2828 . . . . . . . . . . . . . . . . . . . . . . 23 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 ∈ β„‚)
6362subidd 11581 . . . . . . . . . . . . . . . . . . . . . 22 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (𝑧 βˆ’ 𝑧) = 0)
6463negeqd 11476 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = -0)
6562, 62negsubdid 11608 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -(𝑧 βˆ’ 𝑧) = (-𝑧 + 𝑧))
66 neg0 11528 . . . . . . . . . . . . . . . . . . . . . 22 -0 = 0
6766a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = 0)
6864, 65, 673eqtr3d 2775 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-𝑧 + 𝑧) = 0)
6961, 68eqtr3d 2769 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ (-π‘₯ + (i Β· 𝑦)) = 0)
7050renegcld 11663 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -π‘₯ ∈ ℝ)
71 creq0 32501 . . . . . . . . . . . . . . . . . . . 20 ((-π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((-π‘₯ = 0 ∧ 𝑦 = 0) ↔ (-π‘₯ + (i Β· 𝑦)) = 0))
7270, 56, 71syl2anc 583 . . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . 17 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -π‘₯ = 0)
7551, 74negcon1ad 11588 . . . . . . . . . . . . . . . 16 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ -0 = π‘₯)
7653, 75, 673eqtr2d 2773 . . . . . . . . . . . . . . 15 (((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i))) β†’ 𝑧 = 0)
7776ex 412 . . . . . . . . . . . . . 14 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) β†’ 𝑧 = 0))
7877rexlimivv 3194 . . . . . . . . . . . . 13 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ (𝑧 = (π‘₯ Β· 1) ∧ 𝑧 = (𝑦 Β· i)) β†’ 𝑧 = 0)
79 0red 11239 . . . . . . . . . . . . . 14 (𝑧 = 0 β†’ 0 ∈ ℝ)
80 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ π‘₯ = 0)
8180oveq1d 7429 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ π‘₯ = 0) β†’ (π‘₯ Β· 1) = (0 Β· 1))
8281eqeq2d 2738 . . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 18 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ 𝑦 = 0)
8685oveq1d 7429 . . . . . . . . . . . . . . . . 17 ((𝑧 = 0 ∧ 𝑦 = 0) β†’ (𝑦 Β· i) = (0 Β· i))
8786eqeq2d 2738 . . . . . . . . . . . . . . . 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 11425 . . . . . . . . . . . . . . . . . 18 (0 Β· 1) = 0
9089eqeq2i 2740 . . . . . . . . . . . . . . . . 17 (𝑧 = (0 Β· 1) ↔ 𝑧 = 0)
9190biimpri 227 . . . . . . . . . . . . . . . 16 (𝑧 = 0 β†’ 𝑧 = (0 Β· 1))
9228mul02i 11425 . . . . . . . . . . . . . . . . . 18 (0 Β· i) = 0
9392eqeq2i 2740 . . . . . . . . . . . . . . . . 17 (𝑧 = (0 Β· i) ↔ 𝑧 = 0)
9493biimpri 227 . . . . . . . . . . . . . . . 16 (𝑧 = 0 β†’ 𝑧 = (0 Β· i))
9591, 94jca 511 . . . . . . . . . . . . . . 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 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 3960 . . . . . . . . . . 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 4640 . . . . . . . . . . 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 2724 . . . . . . . . 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 33255 . . . . . . 7 (⊀ β†’ ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
106105mptru 1541 . . . . . 6 ({1} βˆͺ {i}) ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
1078, 106eqeltri 2824 . . . . 5 {1, i} ∈ (LIndSβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
108 cnfldadd 21272 . . . . . . . . . 10 + = (+gβ€˜β„‚fld)
10910, 16sraaddg 21054 . . . . . . . . . . 11 (⊀ β†’ (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„)))
110109mptru 1541 . . . . . . . . . 10 (+gβ€˜β„‚fld) = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
111108, 110eqtri 2755 . . . . . . . . 9 + = (+gβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
11234a1i 11 . . . . . . . . 9 (⊀ β†’ ((subringAlg β€˜β„‚fld)β€˜β„) ∈ LMod)
113 1cnd 11231 . . . . . . . . 9 (⊀ β†’ 1 ∈ β„‚)
11428a1i 11 . . . . . . . . 9 (⊀ β†’ i ∈ β„‚)
11524, 111, 38, 4, 42, 9, 112, 113, 114lspprel 20968 . . . . . . . 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 1541 . . . . . . 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 482 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ π‘₯ ∈ ℝ)
118117recnd 11264 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ π‘₯ ∈ β„‚)
119 1cnd 11231 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 1 ∈ β„‚)
120118, 119mulcld 11256 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (π‘₯ Β· 1) ∈ β„‚)
121 simpr 484 . . . . . . . . . . . . 13 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ ℝ)
122121recnd 11264 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ 𝑦 ∈ β„‚)
12328a1i 11 . . . . . . . . . . . 12 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ i ∈ β„‚)
124122, 123mulcld 11256 . . . . . . . . . . 11 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ (𝑦 Β· i) ∈ β„‚)
125120, 124addcld 11255 . . . . . . . . . 10 ((π‘₯ ∈ ℝ ∧ 𝑦 ∈ ℝ) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) ∈ β„‚)
126 eleq1 2816 . . . . . . . . . 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 3194 . . . . . . . 8 (βˆƒπ‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑧 = ((π‘₯ Β· 1) + (𝑦 Β· i)) β†’ 𝑧 ∈ β„‚)
129 recl 15081 . . . . . . . . 9 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ ℝ)
130 simpr 484 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ π‘₯ = (β„œβ€˜π‘§))
131130oveq1d 7429 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ (π‘₯ Β· 1) = ((β„œβ€˜π‘§) Β· 1))
132131oveq1d 7429 . . . . . . . . . . 11 ((𝑧 ∈ β„‚ ∧ π‘₯ = (β„œβ€˜π‘§)) β†’ ((π‘₯ Β· 1) + (𝑦 Β· i)) = (((β„œβ€˜π‘§) Β· 1) + (𝑦 Β· i)))
133132eqeq2d 2738 . . . . . . . . . 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 3173 . . . . . . . . 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 15082 . . . . . . . . . 10 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ ℝ)
136 simpr 484 . . . . . . . . . . . . 13 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ 𝑦 = (β„‘β€˜π‘§))
137136oveq1d 7429 . . . . . . . . . . . 12 ((𝑧 ∈ β„‚ ∧ 𝑦 = (β„‘β€˜π‘§)) β†’ (𝑦 Β· i) = ((β„‘β€˜π‘§) Β· i))
138137oveq2d 7430 . . . . . . . . . . 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 2738 . . . . . . . . . 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 15087 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ 𝑧 = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
141129recnd 11264 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„œβ€˜π‘§) ∈ β„‚)
142141mulridd 11253 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„œβ€˜π‘§) Β· 1) = (β„œβ€˜π‘§))
143135recnd 11264 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ (β„‘β€˜π‘§) ∈ β„‚)
14428a1i 11 . . . . . . . . . . . . 13 (𝑧 ∈ β„‚ β†’ i ∈ β„‚)
145143, 144mulcomd 11257 . . . . . . . . . . . 12 (𝑧 ∈ β„‚ β†’ ((β„‘β€˜π‘§) Β· i) = (i Β· (β„‘β€˜π‘§)))
146142, 145oveq12d 7432 . . . . . . . . . . 11 (𝑧 ∈ β„‚ β†’ (((β„œβ€˜π‘§) Β· 1) + ((β„‘β€˜π‘§) Β· i)) = ((β„œβ€˜π‘§) + (i Β· (β„‘β€˜π‘§))))
147140, 146eqtr4d 2770 . . . . . . . . . 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 275 . . . . . 6 (𝑧 ∈ ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) ↔ 𝑧 ∈ β„‚)
152151eqriv 2724 . . . . 5 ((LSpanβ€˜((subringAlg β€˜β„‚fld)β€˜β„))β€˜{1, i}) = β„‚
153 eqid 2727 . . . . . 6 (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = (LBasisβ€˜((subringAlg β€˜β„‚fld)β€˜β„))
15424, 153, 9islbs4 21753 . . . . 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 33230 . . . 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 32502 . . . 4 1 β‰  i
159 hashprg 14378 . . . . 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 2755 . 2 (dimβ€˜((subringAlg β€˜β„‚fld)β€˜β„)) = 2
1633, 6, 1623eqtr2i 2761 1 (β„‚fld[:]ℝfld) = 2
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534  βŠ€wtru 1535   ∈ wcel 2099   β‰  wne 2935  βˆƒwrex 3065   βˆͺ cun 3942   ∩ cin 3943   βŠ† wss 3944  {csn 4624  {cpr 4626   class class class wbr 5142  β€˜cfv 6542  (class class class)co 7414  β„‚cc 11128  β„cr 11129  0cc0 11130  1c1 11131  ici 11132   + caddc 11133   Β· cmul 11135   βˆ’ cmin 11466  -cneg 11467  2c2 12289  β™―chash 14313  β„œcre 15068  β„‘cim 15069  Basecbs 17171   β†Ύs cress 17200  +gcplusg 17224  .rcmulr 17225  Scalarcsca 17227   ·𝑠 cvsca 17228  0gc0g 17412  LModclmod 20732  LSpanclspn 20844  LBasisclbs 20948  LVecclvec 20976  subringAlg csra 21045  β„‚fldccnfld 21266  β„fldcrefld 21523  LIndSclinds 21726  dimcldim 33228  /FldExtcfldext 33262  [:]cextdg 33265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-reg 9607  ax-inf2 9656  ax-ac2 10478  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-addf 11209  ax-mulf 11210
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9525  df-r1 9779  df-rank 9780  df-dju 9916  df-card 9954  df-acn 9957  df-ac 10131  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-xnn0 12567  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-starv 17239  df-sca 17240  df-vsca 17241  df-ip 17242  df-tset 17243  df-ple 17244  df-ocomp 17245  df-ds 17246  df-unif 17247  df-0g 17414  df-mre 17557  df-mrc 17558  df-mri 17559  df-acs 17560  df-proset 18278  df-drs 18279  df-poset 18296  df-ipo 18511  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-submnd 18732  df-grp 18884  df-minusg 18885  df-sbg 18886  df-subg 19069  df-cntz 19259  df-lsm 19582  df-cmn 19728  df-abl 19729  df-mgp 20066  df-rng 20084  df-ur 20113  df-ring 20166  df-cring 20167  df-oppr 20262  df-dvdsr 20285  df-unit 20286  df-invr 20316  df-dvr 20329  df-subrng 20472  df-subrg 20497  df-drng 20615  df-field 20616  df-lmod 20734  df-lss 20805  df-lsp 20845  df-lbs 20949  df-lvec 20977  df-sra 21047  df-cnfld 21267  df-refld 21524  df-lindf 21727  df-linds 21728  df-dim 33229  df-fldext 33266  df-extdg 33267
This theorem is referenced by: (None)
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