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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 4579 | . . 3 ⊢ (𝑥 ∈ {1, -1} → (𝑥 = 1 ∨ 𝑥 = -1)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
4 | ax-1cn 10583 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | syl6eqel 2918 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ∈ ℂ) |
6 | id 22 | . . . . 5 ⊢ (𝑥 = -1 → 𝑥 = -1) | |
7 | neg1cn 11739 | . . . . 5 ⊢ -1 ∈ ℂ | |
8 | 6, 7 | syl6eqel 2918 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ∈ ℂ) |
9 | 5, 8 | jaoi 851 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ℂ) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ∈ ℂ) |
11 | ax-1ne0 10594 | . . . . . 6 ⊢ 1 ≠ 0 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑥 = 1 → 1 ≠ 0) |
13 | 3, 12 | eqnetrd 3080 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ≠ 0) |
14 | neg1ne0 11741 | . . . . . 6 ⊢ -1 ≠ 0 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 = -1 → -1 ≠ 0) |
16 | 6, 15 | eqnetrd 3080 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ≠ 0) |
17 | 13, 16 | jaoi 851 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ≠ 0) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ≠ 0) |
19 | elpri 4579 | . . 3 ⊢ (𝑦 ∈ {1, -1} → (𝑦 = 1 ∨ 𝑦 = -1)) | |
20 | oveq12 7154 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
21 | 4 | mulid1i 10633 | . . . . . 6 ⊢ (1 · 1) = 1 |
22 | 1ex 10625 | . . . . . . 7 ⊢ 1 ∈ V | |
23 | 22 | prid1 4690 | . . . . . 6 ⊢ 1 ∈ {1, -1} |
24 | 21, 23 | eqeltri 2906 | . . . . 5 ⊢ (1 · 1) ∈ {1, -1} |
25 | 20, 24 | syl6eqel 2918 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
26 | oveq12 7154 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (-1 · 1)) | |
27 | 7 | mulid1i 10633 | . . . . . 6 ⊢ (-1 · 1) = -1 |
28 | negex 10872 | . . . . . . 7 ⊢ -1 ∈ V | |
29 | 28 | prid2 4691 | . . . . . 6 ⊢ -1 ∈ {1, -1} |
30 | 27, 29 | eqeltri 2906 | . . . . 5 ⊢ (-1 · 1) ∈ {1, -1} |
31 | 26, 30 | syl6eqel 2918 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
32 | oveq12 7154 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (1 · -1)) | |
33 | 7 | mulid2i 10634 | . . . . . 6 ⊢ (1 · -1) = -1 |
34 | 33, 29 | eqeltri 2906 | . . . . 5 ⊢ (1 · -1) ∈ {1, -1} |
35 | 32, 34 | syl6eqel 2918 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
36 | oveq12 7154 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (-1 · -1)) | |
37 | neg1mulneg1e1 11838 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
38 | 37, 23 | eqeltri 2906 | . . . . 5 ⊢ (-1 · -1) ∈ {1, -1} |
39 | 36, 38 | syl6eqel 2918 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
40 | 25, 31, 35, 39 | ccase 1029 | . . 3 ⊢ (((𝑥 = 1 ∨ 𝑥 = -1) ∧ (𝑦 = 1 ∨ 𝑦 = -1)) → (𝑥 · 𝑦) ∈ {1, -1}) |
41 | 2, 19, 40 | syl2an 595 | . 2 ⊢ ((𝑥 ∈ {1, -1} ∧ 𝑦 ∈ {1, -1}) → (𝑥 · 𝑦) ∈ {1, -1}) |
42 | oveq2 7153 | . . . . 5 ⊢ (𝑥 = 1 → (1 / 𝑥) = (1 / 1)) | |
43 | 1div1e1 11318 | . . . . . 6 ⊢ (1 / 1) = 1 | |
44 | 43, 23 | eqeltri 2906 | . . . . 5 ⊢ (1 / 1) ∈ {1, -1} |
45 | 42, 44 | syl6eqel 2918 | . . . 4 ⊢ (𝑥 = 1 → (1 / 𝑥) ∈ {1, -1}) |
46 | oveq2 7153 | . . . . 5 ⊢ (𝑥 = -1 → (1 / 𝑥) = (1 / -1)) | |
47 | divneg2 11352 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
48 | 4, 4, 11, 47 | mp3an 1452 | . . . . . . 7 ⊢ -(1 / 1) = (1 / -1) |
49 | 43 | negeqi 10867 | . . . . . . 7 ⊢ -(1 / 1) = -1 |
50 | 48, 49 | eqtr3i 2843 | . . . . . 6 ⊢ (1 / -1) = -1 |
51 | 50, 29 | eqeltri 2906 | . . . . 5 ⊢ (1 / -1) ∈ {1, -1} |
52 | 46, 51 | syl6eqel 2918 | . . . 4 ⊢ (𝑥 = -1 → (1 / 𝑥) ∈ {1, -1}) |
53 | 45, 52 | jaoi 851 | . . 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 20536 | 1 ⊢ {1, -1} ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∖ cdif 3930 {csn 4557 {cpr 4559 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 -cneg 10859 / cdiv 11285 ↾s cress 16472 SubGrpcsubg 18211 mulGrpcmgp 19168 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-rep 5181 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-tpos 7881 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-div 11286 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-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 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-minusg 18045 df-subg 18214 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-cnfld 20474 |
This theorem is referenced by: cnmsgngrp 20651 psgninv 20654 zrhpsgnmhm 20656 |
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