Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 10790 | . 2 ⊢ 0 ∈ ℂ | |
2 | cnlmod.w | . . . . . 6 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) | |
3 | 2 | cnlmodlem1 23987 | . . . . 5 ⊢ (Base‘𝑊) = ℂ |
4 | 3 | eqcomi 2745 | . . . 4 ⊢ ℂ = (Base‘𝑊) |
5 | 4 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → ℂ = (Base‘𝑊)) |
6 | 2 | cnlmodlem2 23988 | . . . . 5 ⊢ (+g‘𝑊) = + |
7 | 6 | eqcomi 2745 | . . . 4 ⊢ + = (+g‘𝑊) |
8 | 7 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → + = (+g‘𝑊)) |
9 | addcl 10776 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | 9 | 3adant1 1132 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) |
11 | addass 10781 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
12 | 11 | adantl 485 | . . 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 10980 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
15 | 14 | adantl 485 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥) |
16 | negcl 11043 | . . . 4 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
17 | 16 | adantl 485 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → -𝑥 ∈ ℂ) |
18 | id 22 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
19 | 16, 18 | addcomd 10999 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
20 | 19 | adantl 485 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
21 | negid 11090 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
22 | 21 | adantl 485 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 + -𝑥) = 0) |
23 | 20, 22 | eqtrd 2771 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = 0) |
24 | 5, 8, 10, 12, 13, 15, 17, 23 | isgrpd 18343 | . 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 23989 | . . . . 5 ⊢ (Scalar‘𝑊) = ℂfld |
28 | 27 | eqcomi 2745 | . . . 4 ⊢ ℂfld = (Scalar‘𝑊) |
29 | 28 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld = (Scalar‘𝑊)) |
30 | 2 | cnlmod4 23990 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = · |
31 | 30 | eqcomi 2745 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) |
32 | 31 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = ( ·𝑠 ‘𝑊)) |
33 | cnfldbas 20321 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
34 | 33 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂ = (Base‘ℂfld)) |
35 | cnfldadd 20322 | . . . 4 ⊢ + = (+g‘ℂfld) | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → + = (+g‘ℂfld)) |
37 | cnfldmul 20323 | . . . 4 ⊢ · = (.r‘ℂfld) | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = (.r‘ℂfld)) |
39 | cnfld1 20342 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → 1 = (1r‘ℂfld)) |
41 | cnring 20339 | . . . 4 ⊢ ℂfld ∈ Ring | |
42 | 41 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld ∈ Ring) |
43 | id 22 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Grp) | |
44 | mulcl 10778 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
45 | 44 | 3adant1 1132 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
46 | adddi 10783 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | |
47 | 46 | adantl 485 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
48 | adddir 10789 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | |
49 | 48 | adantl 485 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
50 | mulass 10782 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | |
51 | 50 | adantl 485 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
52 | mulid2 10797 | . . . 4 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
53 | 52 | adantl 485 | . . 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 19859 | . 2 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ LMod) |
55 | 1, 24, 54 | mp2b 10 | 1 ⊢ 𝑊 ∈ LMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∪ cun 3851 {cpr 4529 〈cop 4533 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 -cneg 11028 ndxcnx 16663 Basecbs 16666 +gcplusg 16749 .rcmulr 16750 Scalarcsca 16752 ·𝑠 cvsca 16753 Grpcgrp 18319 1rcur 19470 Ringcrg 19516 LModclmod 19853 ℂfldccnfld 20317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-cmn 19126 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-lmod 19855 df-cnfld 20318 |
This theorem is referenced by: cnstrcvs 23992 |
Copyright terms: Public domain | W3C validator |