ccfldextdgrr - Mathbox for Thierry Arnoux

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Theorem ccfldextdgrr 33713

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 33688	. . 3 ⊢ ℂ_fld/_FldExtℝ_fld
2		extdgval 33695	. . 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 21566	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
5	4	fveq2i 6879	. . 3 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld))
6	5	fveq2i 6879	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (dim‘((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld)))
7		ccfldsrarelvec 33712	. . . 4 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec
8		df-pr 4604	. . . . . 6 ⊢ {1, i} = ({1} ∪ {i})
9		eqid 2735	. . . . . . . 8 ⊢ (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))
10		eqidd 2736	. . . . . . . . . 10 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘ℝ))
11		cnfld0 21355	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
12	11	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 0 = (0_g‘ℂ_fld))
13		ax-resscn 11186	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14		cnfldbas 21319	. . . . . . . . . . . 12 ⊢ ℂ = (Base‘ℂ_fld)
15	13, 14	sseqtri 4007	. . . . . . . . . . 11 ⊢ ℝ ⊆ (Base‘ℂ_fld)
16	15	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ ⊆ (Base‘ℂ_fld))
17	10, 12, 16	sralmod0 21146	. . . . . . . . 9 ⊢ (⊤ → 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
18	17	mptru 1547	. . . . . . . 8 ⊢ 0 = (0_g‘((subringAlg ‘ℂ_fld)‘ℝ))
19	7	a1i 11	. . . . . . . 8 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec)
20		ax-1cn 11187	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		ax-1ne0 11198	. . . . . . . . . 10 ⊢ 1 ≠ 0
22	10, 16	srabase 21135	. . . . . . . . . . . . 13 ⊢ (⊤ → (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ)))
23	22	mptru 1547	. . . . . . . . . . . 12 ⊢ (Base‘ℂ_fld) = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
24	14, 23	eqtri 2758	. . . . . . . . . . 11 ⊢ ℂ = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
25	24, 18	lindssn 33393	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
26	7, 20, 21, 25	mp3an 1463	. . . . . . . . 9 ⊢ {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
27	26	a1i 11	. . . . . . . 8 ⊢ (⊤ → {1} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
28		ax-icn 11188	. . . . . . . . . 10 ⊢ i ∈ ℂ
29		ine0 11672	. . . . . . . . . 10 ⊢ i ≠ 0
30	24, 18	lindssn 33393	. . . . . . . . . 10 ⊢ ((((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec ∧ i ∈ ℂ ∧ i ≠ 0) → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
31	7, 28, 29, 30	mp3an 1463	. . . . . . . . 9 ⊢ {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
32	31	a1i 11	. . . . . . . 8 ⊢ (⊤ → {i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
33		lveclmod 21064	. . . . . . . . . . . . . . 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 21565	. . . . . . . . . . . . . . . 16 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
36	10, 16	srasca 21138	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ)))
37	36	mptru 1547	. . . . . . . . . . . . . . . 16 ⊢ (ℂ_fld ↾_s ℝ) = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
38	35, 37	eqtri 2758	. . . . . . . . . . . . . . 15 ⊢ ℝ_fld = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
39		cnfldmul 21323	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘ℂ_fld)
40	10, 16	sravsca 21139	. . . . . . . . . . . . . . . . 17 ⊢ (⊤ → (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ)))
41	40	mptru 1547	. . . . . . . . . . . . . . . 16 ⊢ (._r‘ℂ_fld) = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
42	39, 41	eqtri 2758	. . . . . . . . . . . . . . 15 ⊢ · = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
43	38, 4, 24, 42, 9	ellspsn 20960	. . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ↔ ∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1))
45	38, 4, 24, 42, 9	ellspsn 20960	. . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 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 3213	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ (∃𝑥 ∈ ℝ 𝑧 = (𝑥 · 1) ∧ ∃𝑦 ∈ ℝ 𝑧 = (𝑦 · i)))
49		simprl 770	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑥 · 1))
50		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℝ)
51	50	recnd 11263	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℂ)
52	51	mulridd 11252	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑥 · 1) = 𝑥)
53	49, 52	eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 𝑥)
54	53	negeqd 11476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑧 = -𝑥)
55		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (𝑦 · i))
56		simplr 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑦 ∈ ℝ)
57	56	recnd 11263	. . . . . . . . . . . . . . . . . . . . . . 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 11256	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑦 · i) = (i · 𝑦))
60	55, 59	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (i · 𝑦))
61	54, 60	oveq12d 7423	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = (-𝑥 + (i · 𝑦)))
62	53, 51	eqeltrd 2834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 ∈ ℂ)
63	62	subidd 11582	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑧 − 𝑧) = 0)
64	63	negeqd 11476	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = -0)
65	62, 62	negsubdid 11609	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = (-𝑧 + 𝑧))
66		neg0 11529	. . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = 0)
69	61, 68	eqtr3d 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 + (i · 𝑦)) = 0)
70	50	renegcld 11664	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 ∈ ℝ)
71		creq0 32713	. . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 = 0 ∧ 𝑦 = 0))
74	73	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 = 0)
75	51, 74	negcon1ad 11589	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -0 = 𝑥)
76	53, 75, 67	3eqtr2d 2776	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 0)
77	76	ex 412	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0))
78	77	rexlimivv 3186	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0)
79		0red 11238	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → 0 ∈ ℝ)
80		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → 𝑥 = 0)
81	80	oveq1d 7420	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑥 · 1) = (0 · 1))
82	81	eqeq2d 2746	. . . . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . . 14 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i))))
85		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → 𝑦 = 0)
86	85	oveq1d 7420	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑦 · i) = (0 · i))
87	86	eqeq2d 2746	. . . . . . . . . . . . . . . 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 11424	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · 1) = 0
90	89	eqeq2i 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · 1) ↔ 𝑧 = 0)
91	90	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · 1))
92	28	mul02i 11424	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · i) = 0
93	92	eqeq2i 2748	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · i) ↔ 𝑧 = 0)
94	93	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · i))
95	91, 94	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 0 → (𝑧 = (0 · 1) ∧ 𝑧 = (0 · i)))
96	79, 88, 95	rspcedvd 3603	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)))
97	79, 84, 96	rspcedvd 3603	. . . . . . . . . . . . 13 ⊢ (𝑧 = 0 → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)))
98	78, 97	impbii 209	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) ↔ 𝑧 = 0)
99	47, 48, 98	3bitr2i 299	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∧ 𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ 𝑧 = 0)
100		elin 3942	. . . . . . . . . . 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 4617	. . . . . . . . . . 11 ⊢ (𝑧 ∈ {0} ↔ 𝑧 = 0)
102	99, 100, 101	3bitr4i 303	. . . . . . . . . 10 ⊢ (𝑧 ∈ (((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1}) ∩ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{i})) ↔ 𝑧 ∈ {0})
103	102	eqriv 2732	. . . . . . . . 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 33665	. . . . . . 7 ⊢ (⊤ → ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
106	105	mptru 1547	. . . . . 6 ⊢ ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
107	8, 106	eqeltri 2830	. . . . 5 ⊢ {1, i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
108		cnfldadd 21321	. . . . . . . . . 10 ⊢ + = (+_g‘ℂ_fld)
109	10, 16	sraaddg 21136	. . . . . . . . . . 11 ⊢ (⊤ → (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ)))
110	109	mptru 1547	. . . . . . . . . 10 ⊢ (+_g‘ℂ_fld) = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
111	108, 110	eqtri 2758	. . . . . . . . 9 ⊢ + = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
112	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod)
113		1cnd 11230	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
114	28	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
115	24, 111, 38, 4, 42, 9, 112, 113, 114	lspprel 21052	. . . . . . . 8 ⊢ (⊤ → (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i))))
116	115	mptru 1547	. . . . . . 7 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)))
117		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ)
118	117	recnd 11263	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
119		1cnd 11230	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
120	118, 119	mulcld 11255	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 1) ∈ ℂ)
121		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
122	121	recnd 11263	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
123	28	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
124	122, 123	mulcld 11255	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · i) ∈ ℂ)
125	120, 124	addcld 11254	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ)
126		eleq1 2822	. . . . . . . . . 10 ⊢ (𝑧 = ((𝑥 · 1) + (𝑦 · i)) → (𝑧 ∈ ℂ ↔ ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ))
127	125, 126	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ))
128	127	rexlimivv 3186	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ)
129		recl 15129	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ)
130		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → 𝑥 = (ℜ‘𝑧))
131	130	oveq1d 7420	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑥 · 1) = ((ℜ‘𝑧) · 1))
132	131	oveq1d 7420	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → ((𝑥 · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
133	132	eqeq2d 2746	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
134	133	rexbidv 3164	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
135		imcl 15130	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ)
136		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → 𝑦 = (ℑ‘𝑧))
137	136	oveq1d 7420	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑦 · i) = ((ℑ‘𝑧) · i))
138	137	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (((ℜ‘𝑧) · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
139	138	eqeq2d 2746	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i))))
140		replim 15135	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
141	129	recnd 11263	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℂ)
142	141	mulridd 11252	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧) · 1) = (ℜ‘𝑧))
143	135	recnd 11263	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℂ)
144	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → i ∈ ℂ)
145	143, 144	mulcomd 11256	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℑ‘𝑧) · i) = (i · (ℑ‘𝑧)))
146	142, 145	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
147	140, 146	eqtr4d 2773	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
148	135, 139, 147	rspcedvd 3603	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
149	129, 134, 148	rspcedvd 3603	. . . . . . . 8 ⊢ (𝑧 ∈ ℂ → ∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)))
150	128, 149	impbii 209	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 ∈ ℂ)
151	116, 150	bitri 275	. . . . . 6 ⊢ (𝑧 ∈ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) ↔ 𝑧 ∈ ℂ)
152	151	eqriv 2732	. . . . 5 ⊢ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) = ℂ
153		eqid 2735	. . . . . 6 ⊢ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
154	24, 153, 9	islbs4 21792	. . . . 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 711	. . . 4 ⊢ {1, i} ∈ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
156	153	dimval 33640	. . . 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 692	. . 3 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (♯‘{1, i})
158		1nei 32714	. . . 4 ⊢ 1 ≠ i
159		hashprg 14413	. . . . 5 ⊢ ((1 ∈ ℂ ∧ i ∈ ℂ) → (1 ≠ i ↔ (♯‘{1, i}) = 2))
160	20, 28, 159	mp2an 692	. . . 4 ⊢ (1 ≠ i ↔ (♯‘{1, i}) = 2)
161	158, 160	mpbi 230	. . 3 ⊢ (♯‘{1, i}) = 2
162	157, 161	eqtri 2758	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = 2
163	3, 6, 162	3eqtr2i 2764	1 ⊢ (ℂ_fld[:]ℝ_fld) = 2

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 {csn 4601 {cpr 4603 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 ici 11131 + caddc 11132 · cmul 11134 − cmin 11466 -cneg 11467 2c2 12295 ♯chash 14348 ℜcre 15116 ℑcim 15117 Basecbs 17228 ↾_s cress 17251 +_gcplusg 17271 ._rcmulr 17272 Scalarcsca 17274 ·_𝑠 cvsca 17275 0_gc0g 17453 LModclmod 20817 LSpanclspn 20928 LBasisclbs 21032 LVecclvec 21060 subringAlg csra 21129 ℂ_fldccnfld 21315 ℝ_fldcrefld 21564 LIndSclinds 21765 dimcldim 33638 /_FldExtcfldext 33678 [:]cextdg 33681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-reg 9606 ax-inf2 9655 ax-ac2 10477 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-addf 11208 ax-mulf 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-oi 9524 df-r1 9778 df-rank 9779 df-dju 9915 df-card 9953 df-acn 9956 df-ac 10130 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ocomp 17292 df-ds 17293 df-unif 17294 df-0g 17455 df-mre 17598 df-mrc 17599 df-mri 17600 df-acs 17601 df-proset 18306 df-drs 18307 df-poset 18325 df-ipo 18538 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-cring 20196 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-subrng 20506 df-subrg 20530 df-drng 20691 df-field 20692 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lbs 21033 df-lvec 21061 df-sra 21131 df-cnfld 21316 df-refld 21565 df-lindf 21766 df-linds 21767 df-dim 33639 df-fldext 33682 df-extdg 33683

This theorem is referenced by: (None)

W3C validator