![]() |
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 11251 | . 2 ⊢ 0 ∈ ℂ | |
2 | cnlmod.w | . . . . . 6 ⊢ 𝑊 = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} ∪ {〈(Scalar‘ndx), ℂfld〉, 〈( ·𝑠 ‘ndx), · 〉}) | |
3 | 2 | cnlmodlem1 25183 | . . . . 5 ⊢ (Base‘𝑊) = ℂ |
4 | 3 | eqcomi 2744 | . . . 4 ⊢ ℂ = (Base‘𝑊) |
5 | 4 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → ℂ = (Base‘𝑊)) |
6 | 2 | cnlmodlem2 25184 | . . . . 5 ⊢ (+g‘𝑊) = + |
7 | 6 | eqcomi 2744 | . . . 4 ⊢ + = (+g‘𝑊) |
8 | 7 | a1i 11 | . . 3 ⊢ (0 ∈ ℂ → + = (+g‘𝑊)) |
9 | addcl 11235 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
10 | 9 | 3adant1 1129 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) |
11 | addass 11240 | . . . 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 | addlid 11442 | . . . 4 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
15 | 14 | adantl 481 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0 + 𝑥) = 𝑥) |
16 | negcl 11506 | . . . 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 11461 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
20 | 19 | adantl 481 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
21 | negid 11554 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
22 | 21 | adantl 481 | . . . 4 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑥 + -𝑥) = 0) |
23 | 20, 22 | eqtrd 2775 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = 0) |
24 | 5, 8, 10, 12, 13, 15, 17, 23 | isgrpd 18989 | . 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 25185 | . . . . 5 ⊢ (Scalar‘𝑊) = ℂfld |
28 | 27 | eqcomi 2744 | . . . 4 ⊢ ℂfld = (Scalar‘𝑊) |
29 | 28 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld = (Scalar‘𝑊)) |
30 | 2 | cnlmod4 25186 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = · |
31 | 30 | eqcomi 2744 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) |
32 | 31 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = ( ·𝑠 ‘𝑊)) |
33 | cnfldbas 21386 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
34 | 33 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂ = (Base‘ℂfld)) |
35 | cnfldadd 21388 | . . . 4 ⊢ + = (+g‘ℂfld) | |
36 | 35 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → + = (+g‘ℂfld)) |
37 | cnfldmul 21390 | . . . 4 ⊢ · = (.r‘ℂfld) | |
38 | 37 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → · = (.r‘ℂfld)) |
39 | cnfld1 21424 | . . . 4 ⊢ 1 = (1r‘ℂfld) | |
40 | 39 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → 1 = (1r‘ℂfld)) |
41 | cnring 21421 | . . . 4 ⊢ ℂfld ∈ Ring | |
42 | 41 | a1i 11 | . . 3 ⊢ (𝑊 ∈ Grp → ℂfld ∈ Ring) |
43 | id 22 | . . 3 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ Grp) | |
44 | mulcl 11237 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
45 | 44 | 3adant1 1129 | . . 3 ⊢ ((𝑊 ∈ Grp ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
46 | adddi 11242 | . . . 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 11250 | . . . 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 11241 | . . . 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 | mullid 11258 | . . . 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 20881 | . 2 ⊢ (𝑊 ∈ Grp → 𝑊 ∈ LMod) |
55 | 1, 24, 54 | mp2b 10 | 1 ⊢ 𝑊 ∈ LMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {cpr 4633 〈cop 4637 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 -cneg 11491 ndxcnx 17227 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 Scalarcsca 17301 ·𝑠 cvsca 17302 Grpcgrp 18964 1rcur 20199 Ringcrg 20251 LModclmod 20875 ℂfldccnfld 21382 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-cmn 19815 df-mgp 20153 df-ur 20200 df-ring 20253 df-cring 20254 df-lmod 20877 df-cnfld 21383 |
This theorem is referenced by: cnstrcvs 25188 |
Copyright terms: Public domain | W3C validator |