iscvsi - Metamath Proof Explorer

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Theorem iscvsi 25027

Description: Properties that determine a subcomplex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)

**Hypotheses**
Ref	Expression
iscvsp.t	⊢ · = ( ·_𝑠 ‘𝑊)
iscvsp.a	⊢ + = (+_g‘𝑊)
iscvsp.v	⊢ 𝑉 = (Base‘𝑊)
iscvsp.s	⊢ 𝑆 = (Scalar‘𝑊)
iscvsp.k	⊢ 𝐾 = (Base‘𝑆)
iscvsi.1	⊢ 𝑊 ∈ Grp
iscvsi.2	⊢ 𝑆 = (ℂ_fld ↾_s 𝐾)
iscvsi.3	⊢ 𝑆 ∈ DivRing
iscvsi.4	⊢ 𝐾 ∈ (SubRing‘ℂ_fld)
iscvsi.5	⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)
iscvsi.6	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)
iscvsi.7	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
iscvsi.8	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))
iscvsi.9	⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))

**Assertion**
Ref	Expression
iscvsi	⊢ 𝑊 ∈ ℂVec

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

**Proof of Theorem iscvsi**
Step	Hyp	Ref	Expression
1		iscvsi.1	. . 3 ⊢ 𝑊 ∈ Grp
2		iscvsi.3	. . . 4 ⊢ 𝑆 ∈ DivRing
3		iscvsi.2	. . . 4 ⊢ 𝑆 = (ℂ_fld ↾_s 𝐾)
4	2, 3	pm3.2i 470	. . 3 ⊢ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂ_fld ↾_s 𝐾))
5		iscvsi.4	. . 3 ⊢ 𝐾 ∈ (SubRing‘ℂ_fld)
6	1, 4, 5	3pm3.2i 1340	. 2 ⊢ (𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂ_fld ↾_s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂ_fld))
7		iscvsi.5	. . . 4 ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)
8		iscvsi.6	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)
9	8	ancoms 458	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → (𝑦 · 𝑥) ∈ 𝑉)
10		iscvsi.7	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
11	10	3com12 1123	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
12	11	3expa 1118	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
13	12	ralrimiva 3121	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))
14		iscvsi.8	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))
15		iscvsi.9	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))
16	14, 15	jca 511	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
17	16	3comr 1125	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
18	17	3expa 1118	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
19	18	ralrimiva 3121	. . . . . 6 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))
20	9, 13, 19	3jca 1128	. . . . 5 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) → ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))
21	20	ralrimiva 3121	. . . 4 ⊢ (𝑥 ∈ 𝑉 → ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))
22	7, 21	jca 511	. . 3 ⊢ (𝑥 ∈ 𝑉 → ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥))))))
23	22	rgen 3046	. 2 ⊢ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))
24		iscvsp.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑊)
25		iscvsp.a	. . 3 ⊢ + = (+_g‘𝑊)
26		iscvsp.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
27		iscvsp.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑊)
28		iscvsp.k	. . 3 ⊢ 𝐾 = (Base‘𝑆)
29	24, 25, 26, 27, 28	iscvsp 25026	. 2 ⊢ (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂ_fld ↾_s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))
30	6, 23, 29	mpbir2an 711	1 ⊢ 𝑊 ∈ ℂVec

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ‘cfv 6482 (class class class)co 7349 1c1 11010 + caddc 11012 · cmul 11014 Basecbs 17120 ↾_s cress 17141 +_gcplusg 17161 Scalarcsca 17164 ·_𝑠 cvsca 17165 Grpcgrp 18812 SubRingcsubrg 20454 DivRingcdr 20614 ℂ_fldccnfld 21261 ℂVecccvs 25021

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-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-addf 11088 ax-mulf 11089

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-fz 13411 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-0g 17345 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-grp 18815 df-subg 19002 df-cmn 19661 df-mgp 20026 df-ur 20067 df-ring 20120 df-cring 20121 df-subrg 20455 df-lmod 20765 df-lvec 21007 df-cnfld 21262 df-clm 24961 df-cvs 25022

This theorem is referenced by: (None)

W3C validator