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Mirrors > Home > MPE Home > Th. List > cnmsgnsubg | Structured version Visualization version GIF version |
Description: The signs form a multiplicative subgroup of the complex numbers. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
Ref | Expression |
---|---|
cnmsgnsubg.m | ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
Ref | Expression |
---|---|
cnmsgnsubg | ⊢ {1, -1} ∈ (SubGrp‘𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmsgnsubg.m | . 2 ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
2 | elpri 4655 | . . 3 ⊢ (𝑥 ∈ {1, -1} → (𝑥 = 1 ∨ 𝑥 = -1)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
4 | ax-1cn 11212 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | eqeltrdi 2833 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ∈ ℂ) |
6 | id 22 | . . . . 5 ⊢ (𝑥 = -1 → 𝑥 = -1) | |
7 | neg1cn 12373 | . . . . 5 ⊢ -1 ∈ ℂ | |
8 | 6, 7 | eqeltrdi 2833 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ∈ ℂ) |
9 | 5, 8 | jaoi 855 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ℂ) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ∈ ℂ) |
11 | ax-1ne0 11223 | . . . . . 6 ⊢ 1 ≠ 0 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑥 = 1 → 1 ≠ 0) |
13 | 3, 12 | eqnetrd 2997 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ≠ 0) |
14 | neg1ne0 12375 | . . . . . 6 ⊢ -1 ≠ 0 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 = -1 → -1 ≠ 0) |
16 | 6, 15 | eqnetrd 2997 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ≠ 0) |
17 | 13, 16 | jaoi 855 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ≠ 0) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ≠ 0) |
19 | elpri 4655 | . . 3 ⊢ (𝑦 ∈ {1, -1} → (𝑦 = 1 ∨ 𝑦 = -1)) | |
20 | oveq12 7432 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
21 | 4 | mulridi 11264 | . . . . . 6 ⊢ (1 · 1) = 1 |
22 | 1ex 11256 | . . . . . . 7 ⊢ 1 ∈ V | |
23 | 22 | prid1 4770 | . . . . . 6 ⊢ 1 ∈ {1, -1} |
24 | 21, 23 | eqeltri 2821 | . . . . 5 ⊢ (1 · 1) ∈ {1, -1} |
25 | 20, 24 | eqeltrdi 2833 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
26 | oveq12 7432 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (-1 · 1)) | |
27 | 7 | mulridi 11264 | . . . . . 6 ⊢ (-1 · 1) = -1 |
28 | negex 11504 | . . . . . . 7 ⊢ -1 ∈ V | |
29 | 28 | prid2 4771 | . . . . . 6 ⊢ -1 ∈ {1, -1} |
30 | 27, 29 | eqeltri 2821 | . . . . 5 ⊢ (-1 · 1) ∈ {1, -1} |
31 | 26, 30 | eqeltrdi 2833 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
32 | oveq12 7432 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (1 · -1)) | |
33 | 7 | mullidi 11265 | . . . . . 6 ⊢ (1 · -1) = -1 |
34 | 33, 29 | eqeltri 2821 | . . . . 5 ⊢ (1 · -1) ∈ {1, -1} |
35 | 32, 34 | eqeltrdi 2833 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
36 | oveq12 7432 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (-1 · -1)) | |
37 | neg1mulneg1e1 12472 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
38 | 37, 23 | eqeltri 2821 | . . . . 5 ⊢ (-1 · -1) ∈ {1, -1} |
39 | 36, 38 | eqeltrdi 2833 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
40 | 25, 31, 35, 39 | ccase 1035 | . . 3 ⊢ (((𝑥 = 1 ∨ 𝑥 = -1) ∧ (𝑦 = 1 ∨ 𝑦 = -1)) → (𝑥 · 𝑦) ∈ {1, -1}) |
41 | 2, 19, 40 | syl2an 594 | . 2 ⊢ ((𝑥 ∈ {1, -1} ∧ 𝑦 ∈ {1, -1}) → (𝑥 · 𝑦) ∈ {1, -1}) |
42 | oveq2 7431 | . . . . 5 ⊢ (𝑥 = 1 → (1 / 𝑥) = (1 / 1)) | |
43 | 1div1e1 11951 | . . . . . 6 ⊢ (1 / 1) = 1 | |
44 | 43, 23 | eqeltri 2821 | . . . . 5 ⊢ (1 / 1) ∈ {1, -1} |
45 | 42, 44 | eqeltrdi 2833 | . . . 4 ⊢ (𝑥 = 1 → (1 / 𝑥) ∈ {1, -1}) |
46 | oveq2 7431 | . . . . 5 ⊢ (𝑥 = -1 → (1 / 𝑥) = (1 / -1)) | |
47 | divneg2 11985 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
48 | 4, 4, 11, 47 | mp3an 1457 | . . . . . . 7 ⊢ -(1 / 1) = (1 / -1) |
49 | 43 | negeqi 11499 | . . . . . . 7 ⊢ -(1 / 1) = -1 |
50 | 48, 49 | eqtr3i 2755 | . . . . . 6 ⊢ (1 / -1) = -1 |
51 | 50, 29 | eqeltri 2821 | . . . . 5 ⊢ (1 / -1) ∈ {1, -1} |
52 | 46, 51 | eqeltrdi 2833 | . . . 4 ⊢ (𝑥 = -1 → (1 / 𝑥) ∈ {1, -1}) |
53 | 45, 52 | jaoi 855 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → (1 / 𝑥) ∈ {1, -1}) |
54 | 2, 53 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → (1 / 𝑥) ∈ {1, -1}) |
55 | 1, 10, 18, 41, 23, 54 | cnmsubglem 21419 | 1 ⊢ {1, -1} ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∖ cdif 3943 {csn 4632 {cpr 4634 ‘cfv 6553 (class class class)co 7423 ℂcc 11152 0cc0 11154 1c1 11155 · cmul 11159 -cneg 11491 / cdiv 11917 ↾s cress 17237 SubGrpcsubg 19109 mulGrpcmgp 20112 ℂfldccnfld 21335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-addf 11233 ax-mulf 11234 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-fz 13534 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-starv 17276 df-tset 17280 df-ple 17281 df-ds 17283 df-unif 17284 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-minusg 18927 df-subg 19112 df-cmn 19775 df-abl 19776 df-mgp 20113 df-rng 20131 df-ur 20160 df-ring 20213 df-cring 20214 df-oppr 20311 df-dvdsr 20334 df-unit 20335 df-invr 20365 df-dvr 20378 df-drng 20666 df-cnfld 21336 |
This theorem is referenced by: cnmsgngrp 21567 psgninv 21570 zrhpsgnmhm 21572 |
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