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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 9911 | . 2 ⊢ 0 ∈ ℂ | |
2 | cnlmod.w | . . . . . 6 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) | |
3 | 2 | cnlmodlem1 22744 | . . . . 5 ⊢ (Base‘𝑊) = ℂ |
4 | 3 | eqcomi 2619 | . . . 4 ⊢ ℂ = (Base‘𝑊) |
5 | 4 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → ℂ = (Base‘𝑊)) |
6 | 2 | cnlmodlem2 22745 | . . . . 5 ⊢ (+g‘𝑊) = + |
7 | 6 | eqcomi 2619 | . . . 4 ⊢ + = (+g‘𝑊) |
8 | 7 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → + = (+g‘𝑊)) |
9 | addcl 9897 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | 9 | 3adant1 1072 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) |
11 | addass 9902 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
12 | 11 | adantl 481 | . . 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 10098 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
15 | 14 | adantl 481 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥) |
16 | negcl 10160 | . . . 4 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
17 | 16 | adantl 481 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → -𝑥 ∈ ℂ) |
18 | id 22 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → 𝑥 ∈ ℂ) | |
19 | 16, 18 | addcomd 10117 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
20 | 19 | adantl 481 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
21 | negid 10207 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
22 | 21 | adantl 481 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 + -𝑥) = 0) |
23 | 20, 22 | eqtrd 2644 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = 0) |
24 | 5, 8, 10, 12, 13, 15, 17, 23 | isgrpd 17267 | . 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 22746 | . . . . 5 ⊢ (Scalar‘𝑊) = ℂfld |
28 | 27 | eqcomi 2619 | . . . 4 ⊢ ℂfld = (Scalar‘𝑊) |
29 | 28 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld = (Scalar‘𝑊)) |
30 | 2 | cnlmod4 22747 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = · |
31 | 30 | eqcomi 2619 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) |
32 | 31 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = ( ·𝑠 ‘𝑊)) |
33 | cnfldbas 19571 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
34 | 33 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂ = (Base‘ℂfld)) |
35 | cnfldadd 19572 | . . . 4 ⊢ + = (+g‘ℂfld) | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → + = (+g‘ℂfld)) |
37 | cnfldmul 19573 | . . . 4 ⊢ · = (.r‘ℂfld) | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = (.r‘ℂfld)) |
39 | cnfld1 19590 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → 1 = (1r‘ℂfld)) |
41 | cnring 19587 | . . . 4 ⊢ ℂfld ∈ Ring | |
42 | 41 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld ∈ Ring) |
43 | id 22 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Grp) | |
44 | mulcl 9899 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
45 | 44 | 3adant1 1072 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
46 | adddi 9904 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | |
47 | 46 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
48 | adddir 9910 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | |
49 | 48 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
50 | mulass 9903 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | |
51 | 50 | adantl 481 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
52 | mulid2 9917 | . . . 4 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
53 | 52 | adantl 481 | . . 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 18692 | . 2 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ LMod) |
55 | 1, 24, 54 | mp2b 10 | 1 ⊢ 𝑊 ∈ LMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {cpr 4127 〈cop 4131 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 -cneg 10146 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 Grpcgrp 17245 1rcur 18324 Ringcrg 18370 LModclmod 18686 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-cmn 18018 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-lmod 18688 df-cnfld 19568 |
This theorem is referenced by: cnstrcvs 22749 |
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