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Mirrors > Home > MPE Home > Th. List > iscvsi | Structured version Visualization version GIF version |
Description: Properties that determine a subcomplex vector space. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.) |
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 = (ℂfld ↾s 𝐾) |
iscvsi.3 | ⊢ 𝑆 ∈ DivRing |
iscvsi.4 | ⊢ 𝐾 ∈ (SubRing‘ℂfld) |
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))) |
Ref | Expression |
---|---|
iscvsi | ⊢ 𝑊 ∈ ℂVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscvsi.1 | . . 3 ⊢ 𝑊 ∈ Grp | |
2 | iscvsi.3 | . . . 4 ⊢ 𝑆 ∈ DivRing | |
3 | iscvsi.2 | . . . 4 ⊢ 𝑆 = (ℂfld ↾s 𝐾) | |
4 | 2, 3 | pm3.2i 473 | . . 3 ⊢ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂfld ↾s 𝐾)) |
5 | iscvsi.4 | . . 3 ⊢ 𝐾 ∈ (SubRing‘ℂfld) | |
6 | 1, 4, 5 | 3pm3.2i 1335 | . 2 ⊢ (𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂfld ↾s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) |
7 | iscvsi.5 | . . . 4 ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥) | |
8 | iscvsi.6 | . . . . . . 7 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉) | |
9 | 8 | ancoms 461 | . . . . . 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 1119 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) |
12 | 11 | 3expa 1114 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) |
13 | 12 | ralrimiva 3182 | . . . . . 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 514 | . . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))) |
17 | 16 | 3comr 1121 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))) |
18 | 17 | 3expa 1114 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝐾) ∧ 𝑧 ∈ 𝐾) → (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))) |
19 | 18 | ralrimiva 3182 | . . . . . 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 1124 | . . . . 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 3182 | . . . 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 514 | . . 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 3148 | . 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 23726 | . 2 ⊢ (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂfld ↾s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀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 709 | 1 ⊢ 𝑊 ∈ ℂVec |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6349 (class class class)co 7150 1c1 10532 + caddc 10534 · cmul 10536 Basecbs 16477 ↾s cress 16478 +gcplusg 16559 Scalarcsca 16562 ·𝑠 cvsca 16563 Grpcgrp 18097 DivRingcdr 19496 SubRingcsubrg 19525 ℂfldccnfld 20539 ℂVecccvs 23721 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-subg 18270 df-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-lmod 19630 df-lvec 19869 df-cnfld 20540 df-clm 23661 df-cvs 23722 |
This theorem is referenced by: (None) |
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