ccfldextdgrr - Mathbox for Thierry Arnoux

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Theorem ccfldextdgrr 33643

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 33618	. . 3 ⊢ ℂ_fld/_FldExtℝ_fld
2		extdgval 33625	. . 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 21531	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
5	4	fveq2i 6829	. . 3 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld))
6	5	fveq2i 6829	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = (dim‘((subringAlg ‘ℂ_fld)‘(Base‘ℝ_fld)))
7		ccfldsrarelvec 33642	. . . 4 ⊢ ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LVec
8		df-pr 4582	. . . . . 6 ⊢ {1, i} = ({1} ∪ {i})
9		eqid 2729	. . . . . . . 8 ⊢ (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))
10		eqidd 2730	. . . . . . . . . 10 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) = ((subringAlg ‘ℂ_fld)‘ℝ))
11		cnfld0 21317	. . . . . . . . . . 11 ⊢ 0 = (0_g‘ℂ_fld)
12	11	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → 0 = (0_g‘ℂ_fld))
13		ax-resscn 11085	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14		cnfldbas 21283	. . . . . . . . . . . 12 ⊢ ℂ = (Base‘ℂ_fld)
15	13, 14	sseqtri 3986	. . . . . . . . . . 11 ⊢ ℝ ⊆ (Base‘ℂ_fld)
16	15	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → ℝ ⊆ (Base‘ℂ_fld))
17	10, 12, 16	sralmod0 21110	. . . . . . . . 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 11086	. . . . . . . . . 10 ⊢ 1 ∈ ℂ
21		ax-1ne0 11097	. . . . . . . . . 10 ⊢ 1 ≠ 0
22	10, 16	srabase 21099	. . . . . . . . . . . . 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 2752	. . . . . . . . . . 11 ⊢ ℂ = (Base‘((subringAlg ‘ℂ_fld)‘ℝ))
25	24, 18	lindssn 33325	. . . . . . . . . 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 11087	. . . . . . . . . 10 ⊢ i ∈ ℂ
29		ine0 11573	. . . . . . . . . 10 ⊢ i ≠ 0
30	24, 18	lindssn 33325	. . . . . . . . . 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 21028	. . . . . . . . . . . . . . 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 21530	. . . . . . . . . . . . . . . 16 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
36	10, 16	srasca 21102	. . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . 15 ⊢ ℝ_fld = (Scalar‘((subringAlg ‘ℂ_fld)‘ℝ))
39		cnfldmul 21287	. . . . . . . . . . . . . . . 16 ⊢ · = (._r‘ℂ_fld)
40	10, 16	sravsca 21103	. . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . 15 ⊢ · = ( ·_𝑠 ‘((subringAlg ‘ℂ_fld)‘ℝ))
43	38, 4, 24, 42, 9	ellspsn 20924	. . . . . . . . . . . . . 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 20924	. . . . . . . . . . . . . 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 3201	. . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑥 ∈ ℂ)
52	51	mulridd 11151	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑥 · 1) = 𝑥)
53	49, 52	eqtrd 2764	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = 𝑥)
54	53	negeqd 11375	. . . . . . . . . . . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . . . . . . . . . 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 11155	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑦 · i) = (i · 𝑦))
60	55, 59	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 = (i · 𝑦))
61	54, 60	oveq12d 7371	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = (-𝑥 + (i · 𝑦)))
62	53, 51	eqeltrd 2828	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → 𝑧 ∈ ℂ)
63	62	subidd 11481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (𝑧 − 𝑧) = 0)
64	63	negeqd 11375	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = -0)
65	62, 62	negsubdid 11508	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -(𝑧 − 𝑧) = (-𝑧 + 𝑧))
66		neg0 11428	. . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑧 + 𝑧) = 0)
69	61, 68	eqtr3d 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → (-𝑥 + (i · 𝑦)) = 0)
70	50	renegcld 11565	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -𝑥 ∈ ℝ)
71		creq0 32692	. . . . . . . . . . . . . . . . . . . 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 11488	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i))) → -0 = 𝑥)
76	53, 75, 67	3eqtr2d 2770	. . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ (𝑧 = (𝑥 · 1) ∧ 𝑧 = (𝑦 · i)) → 𝑧 = 0)
79		0red 11137	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → 0 ∈ ℝ)
80		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → 𝑥 = 0)
81	80	oveq1d 7368	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑥 = 0) → (𝑥 · 1) = (0 · 1))
82	81	eqeq2d 2740	. . . . . . . . . . . . . . . 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 3153	. . . . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 = 0 ∧ 𝑦 = 0) → (𝑦 · i) = (0 · i))
87	86	eqeq2d 2740	. . . . . . . . . . . . . . . 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 11323	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · 1) = 0
90	89	eqeq2i 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (0 · 1) ↔ 𝑧 = 0)
91	90	biimpri 228	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 0 → 𝑧 = (0 · 1))
92	28	mul02i 11323	. . . . . . . . . . . . . . . . . 18 ⊢ (0 · i) = 0
93	92	eqeq2i 2742	. . . . . . . . . . . . . . . . 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 3581	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 0 → ∃𝑦 ∈ ℝ (𝑧 = (0 · 1) ∧ 𝑧 = (𝑦 · i)))
97	79, 84, 96	rspcedvd 3581	. . . . . . . . . . . . 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 3921	. . . . . . . . . . 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 4595	. . . . . . . . . . 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 2726	. . . . . . . . 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 33597	. . . . . . 7 ⊢ (⊤ → ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ)))
106	105	mptru 1547	. . . . . 6 ⊢ ({1} ∪ {i}) ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
107	8, 106	eqeltri 2824	. . . . 5 ⊢ {1, i} ∈ (LIndS‘((subringAlg ‘ℂ_fld)‘ℝ))
108		cnfldadd 21285	. . . . . . . . . 10 ⊢ + = (+_g‘ℂ_fld)
109	10, 16	sraaddg 21100	. . . . . . . . . . 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 2752	. . . . . . . . 9 ⊢ + = (+_g‘((subringAlg ‘ℂ_fld)‘ℝ))
112	34	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ((subringAlg ‘ℂ_fld)‘ℝ) ∈ LMod)
113		1cnd 11129	. . . . . . . . 9 ⊢ (⊤ → 1 ∈ ℂ)
114	28	a1i 11	. . . . . . . . 9 ⊢ (⊤ → i ∈ ℂ)
115	24, 111, 38, 4, 42, 9, 112, 113, 114	lspprel 21016	. . . . . . . 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 11162	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℂ)
119		1cnd 11129	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
120	118, 119	mulcld 11154	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 1) ∈ ℂ)
121		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
122	121	recnd 11162	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
123	28	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → i ∈ ℂ)
124	122, 123	mulcld 11154	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 · i) ∈ ℂ)
125	120, 124	addcld 11153	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((𝑥 · 1) + (𝑦 · i)) ∈ ℂ)
126		eleq1 2816	. . . . . . . . . 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 3171	. . . . . . . 8 ⊢ (∃𝑥 ∈ ℝ ∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) → 𝑧 ∈ ℂ)
129		recl 15035	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℝ)
130		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → 𝑥 = (ℜ‘𝑧))
131	130	oveq1d 7368	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑥 · 1) = ((ℜ‘𝑧) · 1))
132	131	oveq1d 7368	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → ((𝑥 · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
133	132	eqeq2d 2740	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
134	133	rexbidv 3153	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 = (ℜ‘𝑧)) → (∃𝑦 ∈ ℝ 𝑧 = ((𝑥 · 1) + (𝑦 · i)) ↔ ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i))))
135		imcl 15036	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℝ)
136		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → 𝑦 = (ℑ‘𝑧))
137	136	oveq1d 7368	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑦 · i) = ((ℑ‘𝑧) · i))
138	137	oveq2d 7369	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (((ℜ‘𝑧) · 1) + (𝑦 · i)) = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
139	138	eqeq2d 2740	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 = (ℑ‘𝑧)) → (𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)) ↔ 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i))))
140		replim 15041	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → 𝑧 = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
141	129	recnd 11162	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℜ‘𝑧) ∈ ℂ)
142	141	mulridd 11151	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℜ‘𝑧) · 1) = (ℜ‘𝑧))
143	135	recnd 11162	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → (ℑ‘𝑧) ∈ ℂ)
144	28	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℂ → i ∈ ℂ)
145	143, 144	mulcomd 11155	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℂ → ((ℑ‘𝑧) · i) = (i · (ℑ‘𝑧)))
146	142, 145	oveq12d 7371	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℂ → (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)) = ((ℜ‘𝑧) + (i · (ℑ‘𝑧))))
147	140, 146	eqtr4d 2767	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℂ → 𝑧 = (((ℜ‘𝑧) · 1) + ((ℑ‘𝑧) · i)))
148	135, 139, 147	rspcedvd 3581	. . . . . . . . 9 ⊢ (𝑧 ∈ ℂ → ∃𝑦 ∈ ℝ 𝑧 = (((ℜ‘𝑧) · 1) + (𝑦 · i)))
149	129, 134, 148	rspcedvd 3581	. . . . . . . 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 2726	. . . . 5 ⊢ ((LSpan‘((subringAlg ‘ℂ_fld)‘ℝ))‘{1, i}) = ℂ
153		eqid 2729	. . . . . 6 ⊢ (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ)) = (LBasis‘((subringAlg ‘ℂ_fld)‘ℝ))
154	24, 153, 9	islbs4 21757	. . . . 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 33572	. . . 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 32693	. . . 4 ⊢ 1 ≠ i
159		hashprg 14320	. . . . 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 2752	. 2 ⊢ (dim‘((subringAlg ‘ℂ_fld)‘ℝ)) = 2
163	3, 6, 162	3eqtr2i 2758	1 ⊢ (ℂ_fld[:]ℝ_fld) = 2

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 ∪ cun 3903 ∩ cin 3904 ⊆ wss 3905 {csn 4579 {cpr 4581 class class class wbr 5095 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 ici 11030 + caddc 11031 · cmul 11033 − cmin 11365 -cneg 11366 2c2 12201 ♯chash 14255 ℜcre 15022 ℑcim 15023 Basecbs 17138 ↾_s cress 17159 +_gcplusg 17179 ._rcmulr 17180 Scalarcsca 17182 ·_𝑠 cvsca 17183 0_gc0g 17361 LModclmod 20781 LSpanclspn 20892 LBasisclbs 20996 LVecclvec 21024 subringAlg csra 21093 ℂ_fldccnfld 21279 ℝ_fldcrefld 21529 LIndSclinds 21730 dimcldim 33570 /_FldExtcfldext 33610 [:]cextdg 33612

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-reg 9503 ax-inf2 9556 ax-ac2 10376 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-addf 11107 ax-mulf 11108

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-er 8632 df-map 8762 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-oi 9421 df-r1 9679 df-rank 9680 df-dju 9816 df-card 9854 df-acn 9857 df-ac 10029 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-xnn0 12476 df-z 12490 df-dec 12610 df-uz 12754 df-fz 13429 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ocomp 17200 df-ds 17201 df-unif 17202 df-0g 17363 df-mre 17506 df-mrc 17507 df-mri 17508 df-acs 17509 df-proset 18218 df-drs 18219 df-poset 18237 df-ipo 18452 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-subg 19020 df-cntz 19214 df-lsm 19533 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-dvr 20304 df-subrng 20449 df-subrg 20473 df-drng 20634 df-field 20635 df-lmod 20783 df-lss 20853 df-lsp 20893 df-lbs 20997 df-lvec 21025 df-sra 21095 df-cnfld 21280 df-refld 21530 df-lindf 21731 df-linds 21732 df-dim 33571 df-fldext 33613 df-extdg 33614

This theorem is referenced by: (None)

W3C validator