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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 4145 | . . 3 ⊢ (𝑥 ∈ {1, -1} → (𝑥 = 1 ∨ 𝑥 = -1)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
4 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | syl6eqel 2696 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ∈ ℂ) |
6 | id 22 | . . . . 5 ⊢ (𝑥 = -1 → 𝑥 = -1) | |
7 | neg1cn 11001 | . . . . 5 ⊢ -1 ∈ ℂ | |
8 | 6, 7 | syl6eqel 2696 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ∈ ℂ) |
9 | 5, 8 | jaoi 393 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ℂ) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ∈ ℂ) |
11 | ax-1ne0 9884 | . . . . . 6 ⊢ 1 ≠ 0 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑥 = 1 → 1 ≠ 0) |
13 | 3, 12 | eqnetrd 2849 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ≠ 0) |
14 | neg1ne0 11003 | . . . . . 6 ⊢ -1 ≠ 0 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 = -1 → -1 ≠ 0) |
16 | 6, 15 | eqnetrd 2849 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ≠ 0) |
17 | 13, 16 | jaoi 393 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ≠ 0) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ≠ 0) |
19 | elpri 4145 | . . 3 ⊢ (𝑦 ∈ {1, -1} → (𝑦 = 1 ∨ 𝑦 = -1)) | |
20 | oveq12 6558 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
21 | 4 | mulid1i 9921 | . . . . . 6 ⊢ (1 · 1) = 1 |
22 | 1ex 9914 | . . . . . . 7 ⊢ 1 ∈ V | |
23 | 22 | prid1 4241 | . . . . . 6 ⊢ 1 ∈ {1, -1} |
24 | 21, 23 | eqeltri 2684 | . . . . 5 ⊢ (1 · 1) ∈ {1, -1} |
25 | 20, 24 | syl6eqel 2696 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
26 | oveq12 6558 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (-1 · 1)) | |
27 | 7 | mulid1i 9921 | . . . . . 6 ⊢ (-1 · 1) = -1 |
28 | negex 10158 | . . . . . . 7 ⊢ -1 ∈ V | |
29 | 28 | prid2 4242 | . . . . . 6 ⊢ -1 ∈ {1, -1} |
30 | 27, 29 | eqeltri 2684 | . . . . 5 ⊢ (-1 · 1) ∈ {1, -1} |
31 | 26, 30 | syl6eqel 2696 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
32 | oveq12 6558 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (1 · -1)) | |
33 | 7 | mulid2i 9922 | . . . . . 6 ⊢ (1 · -1) = -1 |
34 | 33, 29 | eqeltri 2684 | . . . . 5 ⊢ (1 · -1) ∈ {1, -1} |
35 | 32, 34 | syl6eqel 2696 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
36 | oveq12 6558 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (-1 · -1)) | |
37 | neg1mulneg1e1 11122 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
38 | 37, 23 | eqeltri 2684 | . . . . 5 ⊢ (-1 · -1) ∈ {1, -1} |
39 | 36, 38 | syl6eqel 2696 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
40 | 25, 31, 35, 39 | ccase 984 | . . 3 ⊢ (((𝑥 = 1 ∨ 𝑥 = -1) ∧ (𝑦 = 1 ∨ 𝑦 = -1)) → (𝑥 · 𝑦) ∈ {1, -1}) |
41 | 2, 19, 40 | syl2an 493 | . 2 ⊢ ((𝑥 ∈ {1, -1} ∧ 𝑦 ∈ {1, -1}) → (𝑥 · 𝑦) ∈ {1, -1}) |
42 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = 1 → (1 / 𝑥) = (1 / 1)) | |
43 | 1div1e1 10596 | . . . . . 6 ⊢ (1 / 1) = 1 | |
44 | 43, 23 | eqeltri 2684 | . . . . 5 ⊢ (1 / 1) ∈ {1, -1} |
45 | 42, 44 | syl6eqel 2696 | . . . 4 ⊢ (𝑥 = 1 → (1 / 𝑥) ∈ {1, -1}) |
46 | oveq2 6557 | . . . . 5 ⊢ (𝑥 = -1 → (1 / 𝑥) = (1 / -1)) | |
47 | divneg2 10628 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
48 | 4, 4, 11, 47 | mp3an 1416 | . . . . . . 7 ⊢ -(1 / 1) = (1 / -1) |
49 | 43 | negeqi 10153 | . . . . . . 7 ⊢ -(1 / 1) = -1 |
50 | 48, 49 | eqtr3i 2634 | . . . . . 6 ⊢ (1 / -1) = -1 |
51 | 50, 29 | eqeltri 2684 | . . . . 5 ⊢ (1 / -1) ∈ {1, -1} |
52 | 46, 51 | syl6eqel 2696 | . . . 4 ⊢ (𝑥 = -1 → (1 / 𝑥) ∈ {1, -1}) |
53 | 45, 52 | jaoi 393 | . . 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 19628 | 1 ⊢ {1, -1} ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 {csn 4125 {cpr 4127 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 -cneg 10146 / cdiv 10563 ↾s cress 15696 SubGrpcsubg 17411 mulGrpcmgp 18312 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-rep 4699 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-tpos 7239 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-div 10564 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-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 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-minusg 17249 df-subg 17414 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-cnfld 19568 |
This theorem is referenced by: cnmsgngrp 19744 psgninv 19747 zrhpsgnmhm 19749 |
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