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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 4654 | . . 3 ⊢ (𝑥 ∈ {1, -1} → (𝑥 = 1 ∨ 𝑥 = -1)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 1 → 𝑥 = 1) | |
4 | ax-1cn 11211 | . . . . 5 ⊢ 1 ∈ ℂ | |
5 | 3, 4 | eqeltrdi 2847 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ∈ ℂ) |
6 | id 22 | . . . . 5 ⊢ (𝑥 = -1 → 𝑥 = -1) | |
7 | neg1cn 12378 | . . . . 5 ⊢ -1 ∈ ℂ | |
8 | 6, 7 | eqeltrdi 2847 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ∈ ℂ) |
9 | 5, 8 | jaoi 857 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ∈ ℂ) |
10 | 2, 9 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ∈ ℂ) |
11 | ax-1ne0 11222 | . . . . . 6 ⊢ 1 ≠ 0 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝑥 = 1 → 1 ≠ 0) |
13 | 3, 12 | eqnetrd 3006 | . . . 4 ⊢ (𝑥 = 1 → 𝑥 ≠ 0) |
14 | neg1ne0 12380 | . . . . . 6 ⊢ -1 ≠ 0 | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝑥 = -1 → -1 ≠ 0) |
16 | 6, 15 | eqnetrd 3006 | . . . 4 ⊢ (𝑥 = -1 → 𝑥 ≠ 0) |
17 | 13, 16 | jaoi 857 | . . 3 ⊢ ((𝑥 = 1 ∨ 𝑥 = -1) → 𝑥 ≠ 0) |
18 | 2, 17 | syl 17 | . 2 ⊢ (𝑥 ∈ {1, -1} → 𝑥 ≠ 0) |
19 | elpri 4654 | . . 3 ⊢ (𝑦 ∈ {1, -1} → (𝑦 = 1 ∨ 𝑦 = -1)) | |
20 | oveq12 7440 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (1 · 1)) | |
21 | 4 | mulridi 11263 | . . . . . 6 ⊢ (1 · 1) = 1 |
22 | 1ex 11255 | . . . . . . 7 ⊢ 1 ∈ V | |
23 | 22 | prid1 4767 | . . . . . 6 ⊢ 1 ∈ {1, -1} |
24 | 21, 23 | eqeltri 2835 | . . . . 5 ⊢ (1 · 1) ∈ {1, -1} |
25 | 20, 24 | eqeltrdi 2847 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
26 | oveq12 7440 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) = (-1 · 1)) | |
27 | 7 | mulridi 11263 | . . . . . 6 ⊢ (-1 · 1) = -1 |
28 | negex 11504 | . . . . . . 7 ⊢ -1 ∈ V | |
29 | 28 | prid2 4768 | . . . . . 6 ⊢ -1 ∈ {1, -1} |
30 | 27, 29 | eqeltri 2835 | . . . . 5 ⊢ (-1 · 1) ∈ {1, -1} |
31 | 26, 30 | eqeltrdi 2847 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = 1) → (𝑥 · 𝑦) ∈ {1, -1}) |
32 | oveq12 7440 | . . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (1 · -1)) | |
33 | 7 | mullidi 11264 | . . . . . 6 ⊢ (1 · -1) = -1 |
34 | 33, 29 | eqeltri 2835 | . . . . 5 ⊢ (1 · -1) ∈ {1, -1} |
35 | 32, 34 | eqeltrdi 2847 | . . . 4 ⊢ ((𝑥 = 1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
36 | oveq12 7440 | . . . . 5 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) = (-1 · -1)) | |
37 | neg1mulneg1e1 12477 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
38 | 37, 23 | eqeltri 2835 | . . . . 5 ⊢ (-1 · -1) ∈ {1, -1} |
39 | 36, 38 | eqeltrdi 2847 | . . . 4 ⊢ ((𝑥 = -1 ∧ 𝑦 = -1) → (𝑥 · 𝑦) ∈ {1, -1}) |
40 | 25, 31, 35, 39 | ccase 1037 | . . 3 ⊢ (((𝑥 = 1 ∨ 𝑥 = -1) ∧ (𝑦 = 1 ∨ 𝑦 = -1)) → (𝑥 · 𝑦) ∈ {1, -1}) |
41 | 2, 19, 40 | syl2an 596 | . 2 ⊢ ((𝑥 ∈ {1, -1} ∧ 𝑦 ∈ {1, -1}) → (𝑥 · 𝑦) ∈ {1, -1}) |
42 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = 1 → (1 / 𝑥) = (1 / 1)) | |
43 | 1div1e1 11956 | . . . . . 6 ⊢ (1 / 1) = 1 | |
44 | 43, 23 | eqeltri 2835 | . . . . 5 ⊢ (1 / 1) ∈ {1, -1} |
45 | 42, 44 | eqeltrdi 2847 | . . . 4 ⊢ (𝑥 = 1 → (1 / 𝑥) ∈ {1, -1}) |
46 | oveq2 7439 | . . . . 5 ⊢ (𝑥 = -1 → (1 / 𝑥) = (1 / -1)) | |
47 | divneg2 11989 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 1 ≠ 0) → -(1 / 1) = (1 / -1)) | |
48 | 4, 4, 11, 47 | mp3an 1460 | . . . . . . 7 ⊢ -(1 / 1) = (1 / -1) |
49 | 43 | negeqi 11499 | . . . . . . 7 ⊢ -(1 / 1) = -1 |
50 | 48, 49 | eqtr3i 2765 | . . . . . 6 ⊢ (1 / -1) = -1 |
51 | 50, 29 | eqeltri 2835 | . . . . 5 ⊢ (1 / -1) ∈ {1, -1} |
52 | 46, 51 | eqeltrdi 2847 | . . . 4 ⊢ (𝑥 = -1 → (1 / 𝑥) ∈ {1, -1}) |
53 | 45, 52 | jaoi 857 | . . 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 21466 | 1 ⊢ {1, -1} ∈ (SubGrp‘𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 {csn 4631 {cpr 4633 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 -cneg 11491 / cdiv 11918 ↾s cress 17274 SubGrpcsubg 19151 mulGrpcmgp 20152 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-rep 5285 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-tpos 8250 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-div 11919 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-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 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-minusg 18968 df-subg 19154 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-cnfld 21383 |
This theorem is referenced by: cnmsgngrp 21615 psgninv 21618 zrhpsgnmhm 21620 |
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