ccfldextdgrr - Mathbox for Thierry Arnoux

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Theorem ccfldextdgrr 33923

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 33897	. . 3 ⊢ ℂ_fld/_FldExtℝ_fld
2		extdgval 33904	. . 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 21631	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
5	4	fveq2i 6859	. . 3 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld))
6	5	fveq2i 6859	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (dim‘((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld)))
7		ccfldsrarelvec 33922	. . . 4 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec
8		df-pr 4579	. . . . . 6 ⊢ {1, i} = ({1} ∪ {i})
9		eqid 2756	. . . . . . . 8 ⊢ (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))
10		eqidd 2757	. . . . . . . . . 10 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘ℝ))
11		cnfld0 21421	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
12	11	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 0 = (0_g‘ℂ_fld))
13		ax-resscn 11120	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14		cnfldbas 21401	. . . . . . . . . . . 12 ⊢ ℂ = (Base‘ℂ_fld)
15	13, 14	sseqtri 3979	. . . . . . . . . . 11 ⊢ ℝ ⊆ (Base‘ℂ_fld)
16	15	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ ⊆ (Base‘ℂ_fld))
17	10, 12, 16	sralmod0 21228	. . . . . . . . 9 ⊢ (⊤ → 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
18	17	mptru 1561	. . . . . . . 8 ⊢ 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ))
19	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec)
20		ax-1cn 11121	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		ax-1ne0 11132	. . . . . . . . . 10 ⊢ 1 ≠ 0
22	10, 16	srabase 21217	. . . . . . . . . . . . 13 ⊢ (⊤ → (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ)))
23	22	mptru 1561	. . . . . . . . . . . 12 ⊢ (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
24	14, 23	eqtri 2779	. . . . . . . . . . 11 ⊢ ℂ = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
25	24, 18	lindssn 33518	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
26	7, 20, 21, 25	mp3an 1476	. . . . . . . . 9 ⊢ {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
27	26	a1i 11	. . . . . . . 8 ⊢ (⊤ → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
28		ax-icn 11122	. . . . . . . . . 10 ⊢ i ∈ ℂ
29		ine0 11612	. . . . . . . . . 10 ⊢ i ≠ 0
30	24, 18	lindssn 33518	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ i ∈ ℂ ∧ i ≠ 0) → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
31	7, 28, 29, 30	mp3an 1476	. . . . . . . . 9 ⊢ {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
32	31	a1i 11	. . . . . . . 8 ⊢ (⊤ → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
33		lveclmod 21146	. . . . . . . . . . . . . . 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 21630	. . . . . . . . . . . . . . . 16 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
36	10, 16	srasca 21220	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ)))
37	36	mptru 1561	. . . . . . . . . . . . . . . 16 ⊢ (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
38	35, 37	eqtri 2779	. . . . . . . . . . . . . . 15 ⊢ ℝ_fld = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
39		cnfldmul 21405	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘ℂ_fld)
40	10, 16	sravsca 21221	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ)))
41	40	mptru 1561	. . . . . . . . . . . . . . . 16 ⊢ (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
42	39, 41	eqtri 2779	. . . . . . . . . . . . . . 15 ⊢ · = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
43	38, 4, 24, 42, 9	ellspsn 21043	. . . . . . . . . . . . . 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 700	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ↔ ∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1))
45	38, 4, 24, 42, 9	ellspsn 21043	. . . . . . . . . . . . . 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 700	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i}) ↔ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i))
47	44, 46	anbi12i 636	. . . . . . . . . . . 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 3228	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1) ∧ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i)))
49		simprl 778	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑥 · 1))
50		simpll 774	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℝ)
51	50	recnd 11200	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℂ)
52	51	mulridd 11189	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑥 · 1) = 𝑥)
53	49, 52	eqtrd 2791	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 𝑥)
54	53	negeqd 11414	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑧 = -𝑥)
55		simprr 780	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑦 · i))
56		simplr 776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑦 ∈ ℝ)
57	56	recnd 11200	. . . . . . . . . . . . . . . . . . . . . . 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 11193	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑦 · i) = (i · 𝑦))
60	55, 59	eqtrd 2791	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (i · 𝑦))
61	54, 60	oveq12d 7403	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = (-𝑥 + (i · 𝑦)))
62	53, 51	eqeltrd 2856	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 ∈ ℂ)
63	62	subidd 11520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑧 − 𝑧) = 0)
64	63	negeqd 11414	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = -0)
65	62, 62	negsubdid 11547	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = (-𝑧 + 𝑧))
66		neg0 11467	. . . . . . . . . . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = 0)
69	61, 68	eqtr3d 2793	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 + (i · 𝑦)) = 0)
70	50	renegcld 11604	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 ∈ ℝ)
71		creq0 32881	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((-𝑥 = 0 ∧ 𝑦 = 0) ↔ (-𝑥 + (i · 𝑦)) = 0))
72	70, 56, 71	syl2anc 592	. . . . . . . . . . . . . . . . . . 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 11527	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -0 = 𝑥)
76	53, 75, 67	3eqtr2d 2797	. . . . . . . . . . . . . . 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 3198	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0)
79		0red 11174	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → 0 ∈ ℝ)
80		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → 𝑥 = 0)
81	80	oveq1d 7400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑥 · 1) = (0 · 1))
82	81	eqeq2d 2767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑧 = (𝑥 · 1) ↔ 𝑧 = (0 · 1)))
83	82	anbi1d 639	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → ((𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i))))
84	83	rexbidv 3180	. . . . . . . . . . . . . 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 7400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑦 · i) = (0 · i))
87	86	eqeq2d 2767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑧 = (𝑦 · i) ↔ 𝑧 = (0 · i)))
88	87	anbi2d 638	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → ((𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (𝑧 = (0 · 1) ∧ 𝑧 = (0 · i))))
89	20	mul02i 11362	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · 1) = 0
90	89	eqeq2i 2769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · 1) ↔ 𝑧 = 0)
91	90	biimpri 230	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · 1))
92	28	mul02i 11362	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · i) = 0
93	92	eqeq2i 2769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · i) ↔ 𝑧 = 0)
94	93	biimpri 230	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · i))
95	91, 94	jca 518	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 0 → (𝑧 = (0 · 1) ∧ 𝑧 = (0 · i)))
96	79, 88, 95	rspcedvd 3578	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)))
97	79, 84, 96	rspcedvd 3578	. . . . . . . . . . . . 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 3915	. . . . . . . . . . 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 4592	. . . . . . . . . . 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 2753	. . . . . . . . 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 33876	. . . . . . 7 ⊢ (⊤ → ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
106	105	mptru 1561	. . . . . 6 ⊢ ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
107	8, 106	eqeltri 2852	. . . . 5 ⊢ {1, i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
108		cnfldadd 21403	. . . . . . . . . 10 ⊢ + = (+_g‘ℂ_fld)
109	10, 16	sraaddg 21218	. . . . . . . . . . 11 ⊢ (⊤ → (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
110	109	mptru 1561	. . . . . . . . . 10 ⊢ (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
111	108, 110	eqtri 2779	. . . . . . . . 9 ⊢ + = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
112	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod)
113		1cnd 11165	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
114	28	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
115	24, 111, 38, 4, 42, 9, 112, 113, 114	lspprel 21134	. . . . . . . 8 ⊢ (⊤ → (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i))))
116	115	mptru 1561	. . . . . . 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 11200	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
119		1cnd 11165	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
120	118, 119	mulcld 11192	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 1) ∈ ℂ)
121		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
122	121	recnd 11200	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
123	28	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
124	122, 123	mulcld 11192	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · i) ∈ ℂ)
125	120, 124	addcld 11191	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ)
126		eleq1 2844	. . . . . . . . . 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 3198	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ)
129		recl 15113	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ)
130		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → 𝑥 = (ℜ‘𝑧))
131	130	oveq1d 7400	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑥 · 1) = ((ℜ‘𝑧) · 1))
132	131	oveq1d 7400	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → ((𝑥 · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
133	132	eqeq2d 2767	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
134	133	rexbidv 3180	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
135		imcl 15114	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ)
136		simpr 487	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → 𝑦 = (ℑ‘𝑧))
137	136	oveq1d 7400	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑦 · i) = ((ℑ‘𝑧) · i))
138	137	oveq2d 7401	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (((ℜ‘𝑧) · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
139	138	eqeq2d 2767	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i))))
140		replim 15119	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
141	129	recnd 11200	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℂ)
142	141	mulridd 11189	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧) · 1) = (ℜ‘𝑧))
143	135	recnd 11200	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℂ)
144	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → i ∈ ℂ)
145	143, 144	mulcomd 11193	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℑ‘𝑧) · i) = (i · (ℑ‘𝑧)))
146	142, 145	oveq12d 7403	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
147	140, 146	eqtr4d 2794	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
148	135, 139, 147	rspcedvd 3578	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
149	129, 134, 148	rspcedvd 3578	. . . . . . . 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 2753	. . . . 5 ⊢ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) = ℂ
153		eqid 2756	. . . . . 6 ⊢ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
154	24, 153, 9	islbs4 21857	. . . . 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 719	. . . 4 ⊢ {1, i} ∈ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
156	153	dimval 33852	. . . 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 700	. . 3 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (♯‘{1, i})
158		1nei 32882	. . . 4 ⊢ 1 ≠ i
159		hashprg 14398	. . . . 5 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ) → (1 ≠ i ↔ (♯‘{1, i}) = 2))
160	20, 28, 159	mp2an 700	. . . 4 ⊢ (1 ≠ i ↔ (♯‘{1, i}) = 2)
161	158, 160	mpbi 232	. . 3 ⊢ (♯‘{1, i}) = 2
162	157, 161	eqtri 2779	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = 2
163	3, 6, 162	3eqtr2i 2785	1 ⊢ (ℂ_fld[:]ℝ_fld) = 2

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1554 ⊤wtru 1555 ∈ wcel 2136 ≠ wne 2951 ∃wrex 3080 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 {csn 4576 {cpr 4578 class class class wbr 5094 ‘cfv 6510 (class class class)co 7385 ℂcc 11061 ℝcr 11062 0cc0 11063 1c1 11064 ici 11065 + caddc 11066 · cmul 11068 − cmin 11404 -cneg 11405 2c2 12262 ♯chash 14333 ℜcre 15100 ℑcim 15101 Basecbs 17221 ↾_s cress 17242 +_gcplusg 17262 ._rcmulr 17263 Scalarcsca 17265 ·_𝑠 cvsca 17266 0_gc0g 17444 LModclmod 20900 LSpanclspn 21011 LBasisclbs 21114 LVecclvec 21142 subringAlg csra 21211 ℂ_fldccnfld 21397 ℝ_fldcrefld 21629 LIndSclinds 21830 dimcldim 33850 /_FldExtcfldext 33889 [:]cextdg 33891

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-reg 9530 ax-inf2 9586 ax-ac2 10410 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-addf 11142 ax-mulf 11143

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-om 7836 df-1st 7959 df-2nd 7960 df-tpos 8194 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-oadd 8429 df-er 8666 df-map 8798 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-oi 9448 df-r1 9712 df-rank 9713 df-dju 9849 df-card 9887 df-acn 9890 df-ac 10062 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-xnn0 12545 df-z 12559 df-dec 12679 df-uz 12830 df-fz 13503 df-hash 14334 df-cj 15102 df-re 15103 df-im 15104 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-starv 17277 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ocomp 17283 df-ds 17284 df-unif 17285 df-0g 17446 df-mre 17590 df-mrc 17591 df-mri 17592 df-acs 17593 df-proset 18302 df-drs 18303 df-poset 18321 df-ipo 18536 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-submnd 18794 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19333 df-lsm 19652 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-ring 20257 df-cring 20258 df-oppr 20358 df-dvdsr 20378 df-unit 20379 df-invr 20409 df-dvr 20422 df-subrng 20568 df-subrg 20592 df-drng 20753 df-field 20754 df-lmod 20902 df-lss 20972 df-lsp 21012 df-lbs 21115 df-lvec 21143 df-sra 21213 df-cnfld 21398 df-refld 21630 df-lindf 21831 df-linds 21832 df-dim 33851 df-fldext 33892 df-extdg 33893

This theorem is referenced by: (None)

W3C validator