Theorem iscvsp 22928

Description: The predicate "is a subcomplex vector space." (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)

Hypotheses

Ref	Expression
iscvsp.t	|- .x. = ( .s ` W )
iscvsp.a	|- .+ = ( +g ` W )
iscvsp.v	|- V = ( Base ` W )
iscvsp.s	|- S = (Scalar ` W )
iscvsp.k	|- K = ( Base ` S )

Assertion

Ref	Expression
iscvsp	|- ( W e. CVec <-> ( ( W e. Grp /\ ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ K e. (SubRing ` ℂfld ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) )

Distinct variable groups:   x, K, y, z   x, S, y, z   x, V, y, z   x, W, y, z   x, .+ , y, z   x, .x. , y, z

Proof of Theorem iscvsp

Step	Hyp	Ref	Expression
1		iscvs 22927	. 2 |- ( W e. CVec <-> ( W e. CMod /\ (Scalar ` W ) e. DivRing ) )
2		iscvsp.t	. . . . 5 |- .x. = ( .s ` W )
3		iscvsp.a	. . . . 5 |- .+ = ( +g ` W )
4		iscvsp.v	. . . . 5 |- V = ( Base ` W )
5		iscvsp.s	. . . . 5 |- S = (Scalar ` W )
6		iscvsp.k	. . . . 5 |- K = ( Base ` S )
7	2, 3, 4, 5, 6	isclmp 22897	. . . 4 |- ( W e. CMod <-> ( ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) )
8	7	anbi2ci 732	. . 3 |- ( ( W e. CMod /\ (Scalar ` W ) e. DivRing ) <-> ( (Scalar ` W ) e. DivRing /\ ( ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) ) )
9		anass 681	. . 3 |- ( ( ( (Scalar ` W ) e. DivRing /\ ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) <-> ( (Scalar ` W ) e. DivRing /\ ( ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) ) )
10		3anan12 1051	. . . . . . 7 |- ( ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) <-> ( S = (ℂfld ↾s K ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) )
11	10	anbi2i 730	. . . . . 6 |- ( ( (Scalar ` W ) e. DivRing /\ ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) <-> ( (Scalar ` W ) e. DivRing /\ ( S = (ℂfld ↾s K ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) ) )
12		anass 681	. . . . . 6 |- ( ( ( (Scalar ` W ) e. DivRing /\ S = (ℂfld ↾s K ) ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) <-> ( (Scalar ` W ) e. DivRing /\ ( S = (ℂfld ↾s K ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) ) )
13	5	eqcomi 2631	. . . . . . . . 9 |- (Scalar ` W ) = S
14	13	eleq1i 2692	. . . . . . . 8 |- ( (Scalar ` W ) e. DivRing <-> S e. DivRing )
15	14	anbi1i 731	. . . . . . 7 |- ( ( (Scalar ` W ) e. DivRing /\ S = (ℂfld ↾s K ) ) <-> ( S e. DivRing /\ S = (ℂfld ↾s K ) ) )
16	15	anbi1i 731	. . . . . 6 |- ( ( ( (Scalar ` W ) e. DivRing /\ S = (ℂfld ↾s K ) ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) <-> ( ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) )
17	11, 12, 16	3bitr2i 288	. . . . 5 |- ( ( (Scalar ` W ) e. DivRing /\ ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) <-> ( ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) )
18		3anan12 1051	. . . . 5 |- ( ( W e. Grp /\ ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ K e. (SubRing ` ℂfld ) ) <-> ( ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ ( W e. Grp /\ K e. (SubRing ` ℂfld ) ) ) )
19	17, 18	bitr4i 267	. . . 4 |- ( ( (Scalar ` W ) e. DivRing /\ ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) <-> ( W e. Grp /\ ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ K e. (SubRing ` ℂfld ) ) )
20	19	anbi1i 731	. . 3 |- ( ( ( (Scalar ` W ) e. DivRing /\ ( W e. Grp /\ S = (ℂfld ↾s K ) /\ K e. (SubRing ` ℂfld ) ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) <-> ( ( W e. Grp /\ ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ K e. (SubRing ` ℂfld ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) )
21	8, 9, 20	3bitr2i 288	. 2 |- ( ( W e. CMod /\ (Scalar ` W ) e. DivRing ) <-> ( ( W e. Grp /\ ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ K e. (SubRing ` ℂfld ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) )
22	1, 21	bitri 264	1 |- ( W e. CVec <-> ( ( W e. Grp /\ ( S e. DivRing /\ S = (ℂfld ↾s K ) ) /\ K e. (SubRing ` ℂfld ) ) /\ A. x e. V ( ( 1 .x. x ) = x /\ A. y e. K ( ( y .x. x ) e. V /\ A. z e. V ( y .x. ( x .+ z ) ) = ( ( y .x. x ) .+ ( y .x. z ) ) /\ A. z e. K ( ( ( z + y ) .x. x ) = ( ( z .x. x ) .+ ( y .x. x ) ) /\ ( ( z x. y ) .x. x ) = ( z .x. ( y .x. x ) ) ) ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   x. cmul 9941   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   Grpcgrp 17422   DivRingcdr 18747  SubRingcsubrg 18776  ℂ_fldccnfld 19746  CModcclm 22862  CVecccvs 22923

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lvec 19103  df-cnfld 19747  df-clm 22863  df-cvs 22924

This theorem is referenced by:  iscvsi  22929