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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 4457 | . . 3 ⊢ (𝑥 ∈ {1, -1} → (𝑥 = 1 ∨ 𝑥 = -1)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
4 | ax-1cn 10385 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | syl6eqel 2868 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ∈ ℂ) |
6 | id 22 | . . . . 5 ⊢ (𝑥 = -1 → 𝑥 = -1) | |
7 | neg1cn 11554 | . . . . 5 ⊢ -1 ∈ ℂ | |
8 | 6, 7 | syl6eqel 2868 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ∈ ℂ) |
9 | 5, 8 | jaoi 843 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ℂ) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ∈ ℂ) |
11 | ax-1ne0 10396 | . . . . . 6 ⊢ 1 ≠ 0 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑥 = 1 → 1 ≠ 0) |
13 | 3, 12 | eqnetrd 3028 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ≠ 0) |
14 | neg1ne0 11556 | . . . . . 6 ⊢ -1 ≠ 0 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 = -1 → -1 ≠ 0) |
16 | 6, 15 | eqnetrd 3028 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ≠ 0) |
17 | 13, 16 | jaoi 843 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ≠ 0) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ≠ 0) |
19 | elpri 4457 | . . 3 ⊢ (𝑦 ∈ {1, -1} → (𝑦 = 1 ∨ 𝑦 = -1)) | |
20 | oveq12 6979 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
21 | 4 | mulid1i 10436 | . . . . . 6 ⊢ (1 · 1) = 1 |
22 | 1ex 10427 | . . . . . . 7 ⊢ 1 ∈ V | |
23 | 22 | prid1 4566 | . . . . . 6 ⊢ 1 ∈ {1, -1} |
24 | 21, 23 | eqeltri 2856 | . . . . 5 ⊢ (1 · 1) ∈ {1, -1} |
25 | 20, 24 | syl6eqel 2868 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
26 | oveq12 6979 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (-1 · 1)) | |
27 | 7 | mulid1i 10436 | . . . . . 6 ⊢ (-1 · 1) = -1 |
28 | negex 10676 | . . . . . . 7 ⊢ -1 ∈ V | |
29 | 28 | prid2 4567 | . . . . . 6 ⊢ -1 ∈ {1, -1} |
30 | 27, 29 | eqeltri 2856 | . . . . 5 ⊢ (-1 · 1) ∈ {1, -1} |
31 | 26, 30 | syl6eqel 2868 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
32 | oveq12 6979 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (1 · -1)) | |
33 | 7 | mulid2i 10437 | . . . . . 6 ⊢ (1 · -1) = -1 |
34 | 33, 29 | eqeltri 2856 | . . . . 5 ⊢ (1 · -1) ∈ {1, -1} |
35 | 32, 34 | syl6eqel 2868 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
36 | oveq12 6979 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (-1 · -1)) | |
37 | neg1mulneg1e1 11653 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
38 | 37, 23 | eqeltri 2856 | . . . . 5 ⊢ (-1 · -1) ∈ {1, -1} |
39 | 36, 38 | syl6eqel 2868 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
40 | 25, 31, 35, 39 | ccase 1018 | . . 3 ⊢ (((𝑥 = 1 ∨ 𝑥 = -1) ∧ (𝑦 = 1 ∨ 𝑦 = -1)) → (𝑥 · 𝑦) ∈ {1, -1}) |
41 | 2, 19, 40 | syl2an 586 | . 2 ⊢ ((𝑥 ∈ {1, -1} ∧ 𝑦 ∈ {1, -1}) → (𝑥 · 𝑦) ∈ {1, -1}) |
42 | oveq2 6978 | . . . . 5 ⊢ (𝑥 = 1 → (1 / 𝑥) = (1 / 1)) | |
43 | 1div1e1 11123 | . . . . . 6 ⊢ (1 / 1) = 1 | |
44 | 43, 23 | eqeltri 2856 | . . . . 5 ⊢ (1 / 1) ∈ {1, -1} |
45 | 42, 44 | syl6eqel 2868 | . . . 4 ⊢ (𝑥 = 1 → (1 / 𝑥) ∈ {1, -1}) |
46 | oveq2 6978 | . . . . 5 ⊢ (𝑥 = -1 → (1 / 𝑥) = (1 / -1)) | |
47 | divneg2 11157 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
48 | 4, 4, 11, 47 | mp3an 1440 | . . . . . . 7 ⊢ -(1 / 1) = (1 / -1) |
49 | 43 | negeqi 10671 | . . . . . . 7 ⊢ -(1 / 1) = -1 |
50 | 48, 49 | eqtr3i 2798 | . . . . . 6 ⊢ (1 / -1) = -1 |
51 | 50, 29 | eqeltri 2856 | . . . . 5 ⊢ (1 / -1) ∈ {1, -1} |
52 | 46, 51 | syl6eqel 2868 | . . . 4 ⊢ (𝑥 = -1 → (1 / 𝑥) ∈ {1, -1}) |
53 | 45, 52 | jaoi 843 | . . 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 20300 | 1 ⊢ {1, -1} ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ∖ cdif 3822 {csn 4435 {cpr 4437 ‘cfv 6182 (class class class)co 6970 ℂcc 10325 0cc0 10327 1c1 10328 · cmul 10332 -cneg 10663 / cdiv 11090 ↾s cress 16330 SubGrpcsubg 18047 mulGrpcmgp 18952 ℂfldccnfld 20237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-addf 10406 ax-mulf 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-tpos 7688 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-fz 12702 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-0g 16561 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-grp 17884 df-minusg 17885 df-subg 18050 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-cring 19013 df-oppr 19086 df-dvdsr 19104 df-unit 19105 df-invr 19135 df-dvr 19146 df-drng 19217 df-cnfld 20238 |
This theorem is referenced by: cnmsgngrp 20415 psgninv 20418 zrhpsgnmhm 20420 |
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