ccfldextdgrr - Mathbox for Thierry Arnoux

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Theorem ccfldextdgrr 30657

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 30638	. . 3 ⊢ ℂ_fld/_FldExtℝ_fld
2		extdgval 30644	. . 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 20436	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
5	4	fveq2i 6548	. . 3 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld))
6	5	fveq2i 6548	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (dim‘((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld)))
7		ccfldsrarelvec 30656	. . . 4 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec
8		df-pr 4481	. . . . . 6 ⊢ {1, i} = ({1} ∪ {i})
9		eqid 2797	. . . . . . . 8 ⊢ (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))
10		eqidd 2798	. . . . . . . . . 10 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘ℝ))
11		cnfld0 20255	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
12	11	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 0 = (0_g‘ℂ_fld))
13		ax-resscn 10447	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14		cnfldbas 20235	. . . . . . . . . . . 12 ⊢ ℂ = (Base‘ℂ_fld)
15	13, 14	sseqtri 3930	. . . . . . . . . . 11 ⊢ ℝ ⊆ (Base‘ℂ_fld)
16	15	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ ⊆ (Base‘ℂ_fld))
17	10, 12, 16	sralmod0 19654	. . . . . . . . 9 ⊢ (⊤ → 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
18	17	mptru 1532	. . . . . . . 8 ⊢ 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ))
19	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec)
20		ax-1cn 10448	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		ax-1ne0 10459	. . . . . . . . . 10 ⊢ 1 ≠ 0
22	10, 16	srabase 19644	. . . . . . . . . . . . 13 ⊢ (⊤ → (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ)))
23	22	mptru 1532	. . . . . . . . . . . 12 ⊢ (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
24	14, 23	eqtri 2821	. . . . . . . . . . 11 ⊢ ℂ = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
25	24, 18	lindssn 30581	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
26	7, 20, 21, 25	mp3an 1453	. . . . . . . . 9 ⊢ {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
27	26	a1i 11	. . . . . . . 8 ⊢ (⊤ → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
28		ax-icn 10449	. . . . . . . . . 10 ⊢ i ∈ ℂ
29		ine0 10929	. . . . . . . . . 10 ⊢ i ≠ 0
30	24, 18	lindssn 30581	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ i ∈ ℂ ∧ i ≠ 0) → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
31	7, 28, 29, 30	mp3an 1453	. . . . . . . . 9 ⊢ {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
32	31	a1i 11	. . . . . . . 8 ⊢ (⊤ → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
33		lveclmod 19572	. . . . . . . . . . . . . . 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 20435	. . . . . . . . . . . . . . . 16 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
36	10, 16	srasca 19647	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ)))
37	36	mptru 1532	. . . . . . . . . . . . . . . 16 ⊢ (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
38	35, 37	eqtri 2821	. . . . . . . . . . . . . . 15 ⊢ ℝ_fld = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
39		cnfldmul 20237	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘ℂ_fld)
40	10, 16	sravsca 19648	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ)))
41	40	mptru 1532	. . . . . . . . . . . . . . . 16 ⊢ (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
42	39, 41	eqtri 2821	. . . . . . . . . . . . . . 15 ⊢ · = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
43	38, 4, 24, 42, 9	lspsnel 19469	. . . . . . . . . . . . . 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 688	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ↔ ∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1))
45	38, 4, 24, 42, 9	lspsnel 19469	. . . . . . . . . . . . . 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 688	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i}) ↔ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i))
47	44, 46	anbi12i 626	. . . . . . . . . . . 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 3330	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1) ∧ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i)))
49		simprl 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑥 · 1))
50		simpll 763	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℝ)
51	50	recnd 10522	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℂ)
52	51	mulid1d 10511	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑥 · 1) = 𝑥)
53	49, 52	eqtrd 2833	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 𝑥)
54	53	negeqd 10733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑧 = -𝑥)
55		simprr 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑦 · i))
56		simplr 765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑦 ∈ ℝ)
57	56	recnd 10522	. . . . . . . . . . . . . . . . . . . . . . 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 10515	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑦 · i) = (i · 𝑦))
60	55, 59	eqtrd 2833	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (i · 𝑦))
61	54, 60	oveq12d 7041	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = (-𝑥 + (i · 𝑦)))
62	53, 51	eqeltrd 2885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 ∈ ℂ)
63	62	subidd 10839	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑧 − 𝑧) = 0)
64	63	negeqd 10733	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = -0)
65	62, 62	negsubdid 10866	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = (-𝑧 + 𝑧))
66		neg0 10786	. . . . . . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = 0)
69	61, 68	eqtr3d 2835	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 + (i · 𝑦)) = 0)
70	50	renegcld 10921	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 ∈ ℝ)
71		creq0 30155	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((-𝑥 = 0 ∧ 𝑦 = 0) ↔ (-𝑥 + (i · 𝑦)) = 0))
72	70, 56, 71	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → ((-𝑥 = 0 ∧ 𝑦 = 0) ↔ (-𝑥 + (i · 𝑦)) = 0))
73	69, 72	mpbird 258	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 = 0 ∧ 𝑦 = 0))
74	73	simpld 495	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 = 0)
75	51, 74	negcon1ad 10846	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -0 = 𝑥)
76	53, 75, 67	3eqtr2d 2839	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 0)
77	76	ex 413	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0))
78	77	rexlimivv 3257	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0)
79		0red 10497	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → 0 ∈ ℝ)
80		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → 𝑥 = 0)
81	80	oveq1d 7038	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑥 · 1) = (0 · 1))
82	81	eqeq2d 2807	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑧 = (𝑥 · 1) ↔ 𝑧 = (0 · 1)))
83	82	anbi1d 629	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → ((𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i))))
84	83	rexbidv 3262	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i))))
85		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → 𝑦 = 0)
86	85	oveq1d 7038	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑦 · i) = (0 · i))
87	86	eqeq2d 2807	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑧 = (𝑦 · i) ↔ 𝑧 = (0 · i)))
88	87	anbi2d 628	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → ((𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (𝑧 = (0 · 1) ∧ 𝑧 = (0 · i))))
89	20	mul02i 10682	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · 1) = 0
90	89	eqeq2i 2809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · 1) ↔ 𝑧 = 0)
91	90	biimpri 229	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · 1))
92	28	mul02i 10682	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · i) = 0
93	92	eqeq2i 2809	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · i) ↔ 𝑧 = 0)
94	93	biimpri 229	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · i))
95	91, 94	jca 512	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 0 → (𝑧 = (0 · 1) ∧ 𝑧 = (0 · i)))
96	79, 88, 95	rspcedvd 3568	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)))
97	79, 84, 96	rspcedvd 3568	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)))
98	78, 97	impbii 210	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ 𝑧 = 0)
99	47, 48, 98	3bitr2i 300	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∧ 𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ 𝑧 = 0)
100		elin 4096	. . . . . . . . . . 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 4494	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {0} ↔ 𝑧 = 0)
102	99, 100, 101	3bitr4i 304	. . . . . . . . . 10 ⊢ (𝑧 ∈ (((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∩ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ 𝑧 ∈ {0})
103	102	eqriv 2794	. . . . . . . . 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 30621	. . . . . . 7 ⊢ (⊤ → ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
106	105	mptru 1532	. . . . . 6 ⊢ ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
107	8, 106	eqeltri 2881	. . . . 5 ⊢ {1, i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
108		cnfldadd 20236	. . . . . . . . . 10 ⊢ + = (+_g‘ℂ_fld)
109	10, 16	sraaddg 19645	. . . . . . . . . . 11 ⊢ (⊤ → (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
110	109	mptru 1532	. . . . . . . . . 10 ⊢ (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
111	108, 110	eqtri 2821	. . . . . . . . 9 ⊢ + = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
112	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod)
113		1cnd 10489	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
114	28	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
115	24, 111, 38, 4, 42, 9, 112, 113, 114	lspprel 19560	. . . . . . . 8 ⊢ (⊤ → (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i))))
116	115	mptru 1532	. . . . . . 7 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)))
117		simpl 483	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
118	117	recnd 10522	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
119		1cnd 10489	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
120	118, 119	mulcld 10514	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 1) ∈ ℂ)
121		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
122	121	recnd 10522	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
123	28	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
124	122, 123	mulcld 10514	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · i) ∈ ℂ)
125	120, 124	addcld 10513	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ)
126		eleq1 2872	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑥 · 1) + (𝑦 · i)) → (𝑧 ∈ ℂ ↔ ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ))
127	125, 126	syl5ibrcom 248	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ))
128	127	rexlimivv 3257	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ)
129		recl 14307	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ)
130		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → 𝑥 = (ℜ‘𝑧))
131	130	oveq1d 7038	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑥 · 1) = ((ℜ‘𝑧) · 1))
132	131	oveq1d 7038	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → ((𝑥 · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
133	132	eqeq2d 2807	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
134	133	rexbidv 3262	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
135		imcl 14308	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ)
136		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → 𝑦 = (ℑ‘𝑧))
137	136	oveq1d 7038	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑦 · i) = ((ℑ‘𝑧) · i))
138	137	oveq2d 7039	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (((ℜ‘𝑧) · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
139	138	eqeq2d 2807	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i))))
140		replim 14313	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
141	129	recnd 10522	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℂ)
142	141	mulid1d 10511	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧) · 1) = (ℜ‘𝑧))
143	135	recnd 10522	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℂ)
144	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → i ∈ ℂ)
145	143, 144	mulcomd 10515	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℑ‘𝑧) · i) = (i · (ℑ‘𝑧)))
146	142, 145	oveq12d 7041	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
147	140, 146	eqtr4d 2836	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
148	135, 139, 147	rspcedvd 3568	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
149	129, 134, 148	rspcedvd 3568	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)))
150	128, 149	impbii 210	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 ∈ ℂ)
151	116, 150	bitri 276	. . . . . 6 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ 𝑧 ∈ ℂ)
152	151	eqriv 2794	. . . . 5 ⊢ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) = ℂ
153		eqid 2797	. . . . . 6 ⊢ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
154	24, 153, 9	islbs4 20662	. . . . 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 707	. . . 4 ⊢ {1, i} ∈ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
156	153	dimval 30601	. . . 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 688	. . 3 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (♯‘{1, i})
158		1nei 30156	. . . 4 ⊢ 1 ≠ i
159		hashprg 13608	. . . . 5 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ) → (1 ≠ i ↔ (♯‘{1, i}) = 2))
160	20, 28, 159	mp2an 688	. . . 4 ⊢ (1 ≠ i ↔ (♯‘{1, i}) = 2)
161	158, 160	mpbi 231	. . 3 ⊢ (♯‘{1, i}) = 2
162	157, 161	eqtri 2821	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = 2
163	3, 6, 162	3eqtr2i 2827	1 ⊢ (ℂ_fld[:]ℝ_fld) = 2

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ⊤wtru 1526 ∈ wcel 2083 ≠ wne 2986 ∃wrex 3108 ∪ cun 3863 ∩ cin 3864 ⊆ wss 3865 {csn 4478 {cpr 4480 class class class wbr 4968 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 0cc0 10390 1c1 10391 ici 10392 + caddc 10393 · cmul 10395 − cmin 10723 -cneg 10724 2c2 11546 ♯chash 13544 ℜcre 14294 ℑcim 14295 Basecbs 16316 ↾_s cress 16317 +_gcplusg 16398 ._rcmulr 16399 Scalarcsca 16401 ·_𝑠 cvsca 16402 0_gc0g 16546 LModclmod 19328 LSpanclspn 19437 LBasisclbs 19540 LVecclvec 19568 subringAlg csra 19634 ℂ_fldccnfld 20231 ℝ_fldcrefld 20434 LIndSclinds 20635 dimcldim 30599 /_FldExtcfldext 30628 [:]cextdg 30631

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-reg 8909 ax-inf2 8957 ax-ac2 9738 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-addf 10469 ax-mulf 10470

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-se 5410 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-isom 6241 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-oi 8827 df-r1 9046 df-rank 9047 df-dju 9183 df-card 9221 df-acn 9224 df-ac 9395 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-xnn0 11822 df-z 11836 df-dec 11953 df-uz 12098 df-fz 12747 df-hash 13545 df-cj 14296 df-re 14297 df-im 14298 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-starv 16413 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ple 16418 df-ocomp 16419 df-ds 16420 df-unif 16421 df-0g 16548 df-mre 16690 df-mrc 16691 df-mri 16692 df-acs 16693 df-proset 17371 df-drs 17372 df-poset 17389 df-ipo 17595 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-submnd 17779 df-grp 17868 df-minusg 17869 df-sbg 17870 df-subg 18034 df-cntz 18192 df-lsm 18495 df-cmn 18639 df-abl 18640 df-mgp 18934 df-ur 18946 df-ring 18993 df-cring 18994 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 df-dvr 19127 df-drng 19198 df-field 19199 df-subrg 19227 df-lmod 19330 df-lss 19398 df-lsp 19438 df-lbs 19541 df-lvec 19569 df-sra 19638 df-cnfld 20232 df-refld 20435 df-lindf 20636 df-linds 20637 df-dim 30600 df-fldext 30632 df-extdg 30633

This theorem is referenced by: (None)

W3C validator