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Mirrors > Home > MPE Home > Th. List > cnlmod | Structured version Visualization version GIF version |
Description: The set of complex numbers is a left module over itself. The vector operation is +, and the scalar product is ·. (Contributed by AV, 20-Sep-2021.) |
Ref | Expression |
---|---|
cnlmod.w | ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) |
Ref | Expression |
---|---|
cnlmod | ⊢ 𝑊 ∈ LMod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10627 | . 2 ⊢ 0 ∈ ℂ | |
2 | cnlmod.w | . . . . . 6 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) | |
3 | 2 | cnlmodlem1 23734 | . . . . 5 ⊢ (Base‘𝑊) = ℂ |
4 | 3 | eqcomi 2830 | . . . 4 ⊢ ℂ = (Base‘𝑊) |
5 | 4 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → ℂ = (Base‘𝑊)) |
6 | 2 | cnlmodlem2 23735 | . . . . 5 ⊢ (+g‘𝑊) = + |
7 | 6 | eqcomi 2830 | . . . 4 ⊢ + = (+g‘𝑊) |
8 | 7 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → + = (+g‘𝑊)) |
9 | addcl 10613 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | 9 | 3adant1 1126 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) |
11 | addass 10618 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
12 | 11 | adantl 484 | . . 3 ⊢ ((0 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
13 | id 22 | . . 3 ⊢ (0 ∈ ℂ → 0 ∈ ℂ) | |
14 | addid2 10817 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
15 | 14 | adantl 484 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥) |
16 | negcl 10880 | . . . 4 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
17 | 16 | adantl 484 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → -𝑥 ∈ ℂ) |
18 | id 22 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
19 | 16, 18 | addcomd 10836 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
20 | 19 | adantl 484 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
21 | negid 10927 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
22 | 21 | adantl 484 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 + -𝑥) = 0) |
23 | 20, 22 | eqtrd 2856 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = 0) |
24 | 5, 8, 10, 12, 13, 15, 17, 23 | isgrpd 18119 | . 2 ⊢ (0 ∈ ℂ → 𝑊 ∈ Grp) |
25 | 4 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂ = (Base‘𝑊)) |
26 | 7 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → + = (+g‘𝑊)) |
27 | 2 | cnlmodlem3 23736 | . . . . 5 ⊢ (Scalar‘𝑊) = ℂfld |
28 | 27 | eqcomi 2830 | . . . 4 ⊢ ℂfld = (Scalar‘𝑊) |
29 | 28 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld = (Scalar‘𝑊)) |
30 | 2 | cnlmod4 23737 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = · |
31 | 30 | eqcomi 2830 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) |
32 | 31 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = ( ·𝑠 ‘𝑊)) |
33 | cnfldbas 20543 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
34 | 33 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂ = (Base‘ℂfld)) |
35 | cnfldadd 20544 | . . . 4 ⊢ + = (+g‘ℂfld) | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → + = (+g‘ℂfld)) |
37 | cnfldmul 20545 | . . . 4 ⊢ · = (.r‘ℂfld) | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = (.r‘ℂfld)) |
39 | cnfld1 20564 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → 1 = (1r‘ℂfld)) |
41 | cnring 20561 | . . . 4 ⊢ ℂfld ∈ Ring | |
42 | 41 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld ∈ Ring) |
43 | id 22 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Grp) | |
44 | mulcl 10615 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
45 | 44 | 3adant1 1126 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
46 | adddi 10620 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | |
47 | 46 | adantl 484 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
48 | adddir 10626 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | |
49 | 48 | adantl 484 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
50 | mulass 10619 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | |
51 | 50 | adantl 484 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
52 | mulid2 10634 | . . . 4 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
53 | 52 | adantl 484 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ ℂ) → (1 · 𝑥) = 𝑥) |
54 | 25, 26, 29, 32, 34, 36, 38, 40, 42, 43, 45, 47, 49, 51, 53 | islmodd 19634 | . 2 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ LMod) |
55 | 1, 24, 54 | mp2b 10 | 1 ⊢ 𝑊 ∈ LMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∪ cun 3934 {cpr 4563 〈cop 4567 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 -cneg 10865 ndxcnx 16474 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 Grpcgrp 18097 1rcur 19245 Ringcrg 19291 LModclmod 19628 ℂfldccnfld 20539 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 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-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 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-cmn 18902 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-lmod 19630 df-cnfld 20540 |
This theorem is referenced by: cnstrcvs 23739 |
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