ccfldextdgrr - Mathbox for Thierry Arnoux

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Theorem ccfldextdgrr 31057

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	⊢ (ℂ_fld[:]ℝ_fld) = 2

Proof of Theorem ccfldextdgrr Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccfldextrr 31038	. . 3 ⊢ ℂ_fld/_FldExtℝ_fld
2		extdgval 31044	. . 3 ⊢ (ℂ_fld/_FldExtℝ_fld → (ℂ_fld[:]ℝ_fld) = (dim‘((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld))))
3	1, 2	ax-mp 5	. 2 ⊢ (ℂ_fld[:]ℝ_fld) = (dim‘((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld)))
4		rebase 20750	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
5	4	fveq2i 6673	. . 3 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld))
6	5	fveq2i 6673	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (dim‘((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld)))
7		ccfldsrarelvec 31056	. . . 4 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec
8		df-pr 4570	. . . . . 6 ⊢ {1, i} = ({1} ∪ {i})
9		eqid 2821	. . . . . . . 8 ⊢ (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))
10		eqidd 2822	. . . . . . . . . 10 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘ℝ))
11		cnfld0 20569	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
12	11	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 0 = (0_g‘ℂ_fld))
13		ax-resscn 10594	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14		cnfldbas 20549	. . . . . . . . . . . 12 ⊢ ℂ = (Base‘ℂ_fld)
15	13, 14	sseqtri 4003	. . . . . . . . . . 11 ⊢ ℝ ⊆ (Base‘ℂ_fld)
16	15	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ ⊆ (Base‘ℂ_fld))
17	10, 12, 16	sralmod0 19960	. . . . . . . . 9 ⊢ (⊤ → 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
18	17	mptru 1544	. . . . . . . 8 ⊢ 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ))
19	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec)
20		ax-1cn 10595	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		ax-1ne0 10606	. . . . . . . . . 10 ⊢ 1 ≠ 0
22	10, 16	srabase 19950	. . . . . . . . . . . . 13 ⊢ (⊤ → (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ)))
23	22	mptru 1544	. . . . . . . . . . . 12 ⊢ (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
24	14, 23	eqtri 2844	. . . . . . . . . . 11 ⊢ ℂ = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
25	24, 18	lindssn 30939	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
26	7, 20, 21, 25	mp3an 1457	. . . . . . . . 9 ⊢ {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
27	26	a1i 11	. . . . . . . 8 ⊢ (⊤ → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
28		ax-icn 10596	. . . . . . . . . 10 ⊢ i ∈ ℂ
29		ine0 11075	. . . . . . . . . 10 ⊢ i ≠ 0
30	24, 18	lindssn 30939	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ i ∈ ℂ ∧ i ≠ 0) → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
31	7, 28, 29, 30	mp3an 1457	. . . . . . . . 9 ⊢ {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
32	31	a1i 11	. . . . . . . 8 ⊢ (⊤ → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
33		lveclmod 19878	. . . . . . . . . . . . . . 15 ⊢ (((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod)
34	7, 33	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod
35		df-refld 20749	. . . . . . . . . . . . . . . 16 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
36	10, 16	srasca 19953	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ)))
37	36	mptru 1544	. . . . . . . . . . . . . . . 16 ⊢ (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
38	35, 37	eqtri 2844	. . . . . . . . . . . . . . 15 ⊢ ℝ_fld = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
39		cnfldmul 20551	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘ℂ_fld)
40	10, 16	sravsca 19954	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ)))
41	40	mptru 1544	. . . . . . . . . . . . . . . 16 ⊢ (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
42	39, 41	eqtri 2844	. . . . . . . . . . . . . . 15 ⊢ · = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
43	38, 4, 24, 42, 9	lspsnel 19775	. . . . . . . . . . . . . 14 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod ∧ 1 ∈ ℂ) → (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ↔ ∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1)))
44	34, 20, 43	mp2an 690	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ↔ ∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1))
45	38, 4, 24, 42, 9	lspsnel 19775	. . . . . . . . . . . . . 14 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod ∧ i ∈ ℂ) → (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i}) ↔ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i)))
46	34, 28, 45	mp2an 690	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i}) ↔ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i))
47	44, 46	anbi12i 628	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∧ 𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ (∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1) ∧ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i)))
48		reeanv 3367	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1) ∧ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i)))
49		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑥 · 1))
50		simpll 765	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℝ)
51	50	recnd 10669	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℂ)
52	51	mulid1d 10658	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑥 · 1) = 𝑥)
53	49, 52	eqtrd 2856	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 𝑥)
54	53	negeqd 10880	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑧 = -𝑥)
55		simprr 771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑦 · i))
56		simplr 767	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑦 ∈ ℝ)
57	56	recnd 10669	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑦 ∈ ℂ)
58	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → i ∈ ℂ)
59	57, 58	mulcomd 10662	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑦 · i) = (i · 𝑦))
60	55, 59	eqtrd 2856	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (i · 𝑦))
61	54, 60	oveq12d 7174	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = (-𝑥 + (i · 𝑦)))
62	53, 51	eqeltrd 2913	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 ∈ ℂ)
63	62	subidd 10985	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑧 − 𝑧) = 0)
64	63	negeqd 10880	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = -0)
65	62, 62	negsubdid 11012	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = (-𝑧 + 𝑧))
66		neg0 10932	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ -0 = 0
67	66	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -0 = 0)
68	64, 65, 67	3eqtr3d 2864	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = 0)
69	61, 68	eqtr3d 2858	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 + (i · 𝑦)) = 0)
70	50	renegcld 11067	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 ∈ ℝ)
71		creq0 30471	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((-𝑥 = 0 ∧ 𝑦 = 0) ↔ (-𝑥 + (i · 𝑦)) = 0))
72	70, 56, 71	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → ((-𝑥 = 0 ∧ 𝑦 = 0) ↔ (-𝑥 + (i · 𝑦)) = 0))
73	69, 72	mpbird 259	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 = 0 ∧ 𝑦 = 0))
74	73	simpld 497	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 = 0)
75	51, 74	negcon1ad 10992	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -0 = 𝑥)
76	53, 75, 67	3eqtr2d 2862	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 0)
77	76	ex 415	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0))
78	77	rexlimivv 3292	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0)
79		0red 10644	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → 0 ∈ ℝ)
80		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → 𝑥 = 0)
81	80	oveq1d 7171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑥 · 1) = (0 · 1))
82	81	eqeq2d 2832	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑧 = (𝑥 · 1) ↔ 𝑧 = (0 · 1)))
83	82	anbi1d 631	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → ((𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i))))
84	83	rexbidv 3297	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i))))
85		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → 𝑦 = 0)
86	85	oveq1d 7171	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑦 · i) = (0 · i))
87	86	eqeq2d 2832	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑧 = (𝑦 · i) ↔ 𝑧 = (0 · i)))
88	87	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → ((𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (𝑧 = (0 · 1) ∧ 𝑧 = (0 · i))))
89	20	mul02i 10829	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · 1) = 0
90	89	eqeq2i 2834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · 1) ↔ 𝑧 = 0)
91	90	biimpri 230	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · 1))
92	28	mul02i 10829	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · i) = 0
93	92	eqeq2i 2834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · i) ↔ 𝑧 = 0)
94	93	biimpri 230	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · i))
95	91, 94	jca 514	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 0 → (𝑧 = (0 · 1) ∧ 𝑧 = (0 · i)))
96	79, 88, 95	rspcedvd 3626	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)))
97	79, 84, 96	rspcedvd 3626	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)))
98	78, 97	impbii 211	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ 𝑧 = 0)
99	47, 48, 98	3bitr2i 301	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∧ 𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ 𝑧 = 0)
100		elin 4169	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∩ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∧ 𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})))
101		velsn 4583	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {0} ↔ 𝑧 = 0)
102	99, 100, 101	3bitr4i 305	. . . . . . . . . 10 ⊢ (𝑧 ∈ (((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∩ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ 𝑧 ∈ {0})
103	102	eqriv 2818	. . . . . . . . 9 ⊢ (((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∩ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) = {0}
104	103	a1i 11	. . . . . . . 8 ⊢ (⊤ → (((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∩ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) = {0})
105	9, 18, 19, 27, 32, 104	lindsun 31021	. . . . . . 7 ⊢ (⊤ → ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
106	105	mptru 1544	. . . . . 6 ⊢ ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
107	8, 106	eqeltri 2909	. . . . 5 ⊢ {1, i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
108		cnfldadd 20550	. . . . . . . . . 10 ⊢ + = (+_g‘ℂ_fld)
109	10, 16	sraaddg 19951	. . . . . . . . . . 11 ⊢ (⊤ → (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
110	109	mptru 1544	. . . . . . . . . 10 ⊢ (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
111	108, 110	eqtri 2844	. . . . . . . . 9 ⊢ + = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
112	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod)
113		1cnd 10636	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
114	28	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
115	24, 111, 38, 4, 42, 9, 112, 113, 114	lspprel 19866	. . . . . . . 8 ⊢ (⊤ → (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i))))
116	115	mptru 1544	. . . . . . 7 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)))
117		simpl 485	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
118	117	recnd 10669	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
119		1cnd 10636	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
120	118, 119	mulcld 10661	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 1) ∈ ℂ)
121		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
122	121	recnd 10669	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
123	28	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
124	122, 123	mulcld 10661	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · i) ∈ ℂ)
125	120, 124	addcld 10660	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ)
126		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑥 · 1) + (𝑦 · i)) → (𝑧 ∈ ℂ ↔ ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ))
127	125, 126	syl5ibrcom 249	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ))
128	127	rexlimivv 3292	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ)
129		recl 14469	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ)
130		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → 𝑥 = (ℜ‘𝑧))
131	130	oveq1d 7171	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑥 · 1) = ((ℜ‘𝑧) · 1))
132	131	oveq1d 7171	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → ((𝑥 · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
133	132	eqeq2d 2832	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
134	133	rexbidv 3297	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
135		imcl 14470	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ)
136		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → 𝑦 = (ℑ‘𝑧))
137	136	oveq1d 7171	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑦 · i) = ((ℑ‘𝑧) · i))
138	137	oveq2d 7172	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (((ℜ‘𝑧) · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
139	138	eqeq2d 2832	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i))))
140		replim 14475	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
141	129	recnd 10669	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℂ)
142	141	mulid1d 10658	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧) · 1) = (ℜ‘𝑧))
143	135	recnd 10669	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℂ)
144	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → i ∈ ℂ)
145	143, 144	mulcomd 10662	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℑ‘𝑧) · i) = (i · (ℑ‘𝑧)))
146	142, 145	oveq12d 7174	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
147	140, 146	eqtr4d 2859	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
148	135, 139, 147	rspcedvd 3626	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
149	129, 134, 148	rspcedvd 3626	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)))
150	128, 149	impbii 211	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 ∈ ℂ)
151	116, 150	bitri 277	. . . . . 6 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ 𝑧 ∈ ℂ)
152	151	eqriv 2818	. . . . 5 ⊢ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) = ℂ
153		eqid 2821	. . . . . 6 ⊢ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
154	24, 153, 9	islbs4 20976	. . . . 5 ⊢ ({1, i} ∈ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ)) ↔ ({1, i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)) ∧ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) = ℂ))
155	107, 152, 154	mpbir2an 709	. . . 4 ⊢ {1, i} ∈ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
156	153	dimval 31001	. . . 4 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ {1, i} ∈ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))) → (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (♯‘{1, i}))
157	7, 155, 156	mp2an 690	. . 3 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (♯‘{1, i})
158		1nei 30472	. . . 4 ⊢ 1 ≠ i
159		hashprg 13757	. . . . 5 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ) → (1 ≠ i ↔ (♯‘{1, i}) = 2))
160	20, 28, 159	mp2an 690	. . . 4 ⊢ (1 ≠ i ↔ (♯‘{1, i}) = 2)
161	158, 160	mpbi 232	. . 3 ⊢ (♯‘{1, i}) = 2
162	157, 161	eqtri 2844	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = 2
163	3, 6, 162	3eqtr2i 2850	1 ⊢ (ℂ_fld[:]ℝ_fld) = 2

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 {csn 4567 {cpr 4569 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 − cmin 10870 -cneg 10871 2c2 11693 ♯chash 13691 ℜcre 14456 ℑcim 14457 Basecbs 16483 ↾_s cress 16484 +_gcplusg 16565 ._rcmulr 16566 Scalarcsca 16568 ·_𝑠 cvsca 16569 0_gc0g 16713 LModclmod 19634 LSpanclspn 19743 LBasisclbs 19846 LVecclvec 19874 subringAlg csra 19940 ℂ_fldccnfld 20545 ℝ_fldcrefld 20748 LIndSclinds 20949 dimcldim 30999 /_FldExtcfldext 31028 [:]cextdg 31031

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-reg 9056 ax-inf2 9104 ax-ac2 9885 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-oi 8974 df-r1 9193 df-rank 9194 df-dju 9330 df-card 9368 df-acn 9371 df-ac 9542 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ocomp 16586 df-ds 16587 df-unif 16588 df-0g 16715 df-mre 16857 df-mrc 16858 df-mri 16859 df-acs 16860 df-proset 17538 df-drs 17539 df-poset 17556 df-ipo 17762 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-field 19505 df-subrg 19533 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lbs 19847 df-lvec 19875 df-sra 19944 df-cnfld 20546 df-refld 20749 df-lindf 20950 df-linds 20951 df-dim 31000 df-fldext 31032 df-extdg 31033

This theorem is referenced by: (None)

W3C validator