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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 10621 | . 2 ⊢ 0 ∈ ℂ | |
2 | cnlmod.w | . . . . . 6 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) | |
3 | 2 | cnlmodlem1 23667 | . . . . 5 ⊢ (Base‘𝑊) = ℂ |
4 | 3 | eqcomi 2827 | . . . 4 ⊢ ℂ = (Base‘𝑊) |
5 | 4 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → ℂ = (Base‘𝑊)) |
6 | 2 | cnlmodlem2 23668 | . . . . 5 ⊢ (+g‘𝑊) = + |
7 | 6 | eqcomi 2827 | . . . 4 ⊢ + = (+g‘𝑊) |
8 | 7 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → + = (+g‘𝑊)) |
9 | addcl 10607 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | 9 | 3adant1 1122 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) |
11 | addass 10612 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
12 | 11 | adantl 482 | . . 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 10811 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
15 | 14 | adantl 482 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥) |
16 | negcl 10874 | . . . 4 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
17 | 16 | adantl 482 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → -𝑥 ∈ ℂ) |
18 | id 22 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
19 | 16, 18 | addcomd 10830 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
20 | 19 | adantl 482 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
21 | negid 10921 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
22 | 21 | adantl 482 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 + -𝑥) = 0) |
23 | 20, 22 | eqtrd 2853 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = 0) |
24 | 5, 8, 10, 12, 13, 15, 17, 23 | isgrpd 18063 | . 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 23669 | . . . . 5 ⊢ (Scalar‘𝑊) = ℂfld |
28 | 27 | eqcomi 2827 | . . . 4 ⊢ ℂfld = (Scalar‘𝑊) |
29 | 28 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld = (Scalar‘𝑊)) |
30 | 2 | cnlmod4 23670 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = · |
31 | 30 | eqcomi 2827 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) |
32 | 31 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = ( ·𝑠 ‘𝑊)) |
33 | cnfldbas 20477 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
34 | 33 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂ = (Base‘ℂfld)) |
35 | cnfldadd 20478 | . . . 4 ⊢ + = (+g‘ℂfld) | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → + = (+g‘ℂfld)) |
37 | cnfldmul 20479 | . . . 4 ⊢ · = (.r‘ℂfld) | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = (.r‘ℂfld)) |
39 | cnfld1 20498 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → 1 = (1r‘ℂfld)) |
41 | cnring 20495 | . . . 4 ⊢ ℂfld ∈ Ring | |
42 | 41 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld ∈ Ring) |
43 | id 22 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Grp) | |
44 | mulcl 10609 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
45 | 44 | 3adant1 1122 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
46 | adddi 10614 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | |
47 | 46 | adantl 482 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
48 | adddir 10620 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | |
49 | 48 | adantl 482 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
50 | mulass 10613 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | |
51 | 50 | adantl 482 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
52 | mulid2 10628 | . . . 4 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
53 | 52 | adantl 482 | . . 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 19569 | . 2 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ LMod) |
55 | 1, 24, 54 | mp2b 10 | 1 ⊢ 𝑊 ∈ LMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 {cpr 4559 〈cop 4563 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 -cneg 10859 ndxcnx 16468 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 Grpcgrp 18041 1rcur 19180 Ringcrg 19226 LModclmod 19563 ℂfldccnfld 20473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-cmn 18837 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-lmod 19565 df-cnfld 20474 |
This theorem is referenced by: cnstrcvs 23672 |
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