vciOLD - Metamath Proof Explorer

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Theorem vciOLD 27416

Description: Obsolete version of cvsi 22930 as of 21-Sep-2021. The properties of a complex vector space, which is an Abelian group (i.e. the vectors, with the operation of vector addition) accompanied by a scalar multiplication operation on the field of complex numbers. The variable

was chosen because

is already used for the universal class. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
vciOLD.1
vciOLD.2
vciOLD.3

**Assertion**
Ref	Expression
vciOLD	$/\$ $/\$ $/\$ $/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

(

)

Proof of Theorem vciOLD Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vciOLD.1	. . . . 5
2	1	eqeq2i 2634	. . . 4
3		eleq1 2689	. . . . 5
4		rneq 5351	. . . . . . 7
5		vciOLD.3	. . . . . . 7
6	4, 5	syl6eqr 2674	. . . . . 6
7		xpeq2 5129	. . . . . . . 8
8	7	feq2d 6031	. . . . . . 7
9		feq3 6028	. . . . . . 7
10	8, 9	bitrd 268	. . . . . 6
11	6, 10	syl 17	. . . . 5
12		oveq 6656	. . . . . . . . . . . 12
13	12	oveq2d 6666	. . . . . . . . . . 11
14		oveq 6656	. . . . . . . . . . 11
15	13, 14	eqeq12d 2637	. . . . . . . . . 10
16	6, 15	raleqbidv 3152	. . . . . . . . 9
17		oveq 6656	. . . . . . . . . . . 12
18	17	eqeq2d 2632	. . . . . . . . . . 11
19	18	anbi1d 741	. . . . . . . . . 10 $/\$ $/\$
20	19	ralbidv 2986	. . . . . . . . 9 $/\$ $/\$
21	16, 20	anbi12d 747	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
22	21	ralbidv 2986	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
23	22	anbi2d 740	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	6, 23	raleqbidv 3152	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	3, 11, 24	3anbi123d 1399	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	2, 25	sylbir 225	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		vciOLD.2	. . . . 5
28	27	eqeq2i 2634	. . . 4
29		feq1 6026	. . . . 5
30		oveq 6656	. . . . . . . 8
31	30	eqeq1d 2624	. . . . . . 7
32		oveq 6656	. . . . . . . . . . 11
33		oveq 6656	. . . . . . . . . . . 12
34		oveq 6656	. . . . . . . . . . . 12
35	33, 34	oveq12d 6668	. . . . . . . . . . 11
36	32, 35	eqeq12d 2637	. . . . . . . . . 10
37	36	ralbidv 2986	. . . . . . . . 9
38		oveq 6656	. . . . . . . . . . . 12
39		oveq 6656	. . . . . . . . . . . . 13
40	33, 39	oveq12d 6668	. . . . . . . . . . . 12
41	38, 40	eqeq12d 2637	. . . . . . . . . . 11
42		oveq 6656	. . . . . . . . . . . 12
43	39	oveq2d 6666	. . . . . . . . . . . . 13
44		oveq 6656	. . . . . . . . . . . . 13
45	43, 44	eqtrd 2656	. . . . . . . . . . . 12
46	42, 45	eqeq12d 2637	. . . . . . . . . . 11
47	41, 46	anbi12d 747	. . . . . . . . . 10 $/\$ $/\$
48	47	ralbidv 2986	. . . . . . . . 9 $/\$ $/\$
49	37, 48	anbi12d 747	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50	49	ralbidv 2986	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	31, 50	anbi12d 747	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	51	ralbidv 2986	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	29, 52	3anbi23d 1402	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	28, 53	sylbir 225	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	26, 54	elopabi 7231	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		df-vc 27414	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
57	55, 56	eleq2s 2719	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wral 2912

copab 4712

cxp 5112

crn 5115

wf 5884

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cc 9934

c1 9937

caddc 9939

cmul 9941

cablo 27398

cvc 27413

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-1st 7168 df-2nd 7169 df-vc 27414

This theorem is referenced by: vcsm 27417 vcidOLD 27419 vcdi 27420 vcdir 27421 vcass 27422 vcablo 27424

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