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Mirrors > Home > MPE Home > Th. List > cnmsgngrp | Structured version Visualization version GIF version |
Description: The group of signs under multiplication. (Contributed by Stefan O'Rear, 28-Aug-2015.) |
Ref | Expression |
---|---|
cnmsgngrp.u | ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) |
Ref | Expression |
---|---|
cnmsgngrp | ⊢ 𝑈 ∈ Grp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
2 | 1 | cnmsgnsubg 20723 | . 2 ⊢ {1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) |
3 | cnmsgngrp.u | . . . 4 ⊢ 𝑈 = ((mulGrp‘ℂfld) ↾s {1, -1}) | |
4 | cnex 10620 | . . . . . 6 ⊢ ℂ ∈ V | |
5 | 4 | difexi 5234 | . . . . 5 ⊢ (ℂ ∖ {0}) ∈ V |
6 | ax-1cn 10597 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
7 | ax-1ne0 10608 | . . . . . . 7 ⊢ 1 ≠ 0 | |
8 | eldifsn 4721 | . . . . . . 7 ⊢ (1 ∈ (ℂ ∖ {0}) ↔ (1 ∈ ℂ ∧ 1 ≠ 0)) | |
9 | 6, 7, 8 | mpbir2an 709 | . . . . . 6 ⊢ 1 ∈ (ℂ ∖ {0}) |
10 | neg1cn 11754 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
11 | neg1ne0 11756 | . . . . . . 7 ⊢ -1 ≠ 0 | |
12 | eldifsn 4721 | . . . . . . 7 ⊢ (-1 ∈ (ℂ ∖ {0}) ↔ (-1 ∈ ℂ ∧ -1 ≠ 0)) | |
13 | 10, 11, 12 | mpbir2an 709 | . . . . . 6 ⊢ -1 ∈ (ℂ ∖ {0}) |
14 | prssi 4756 | . . . . . 6 ⊢ ((1 ∈ (ℂ ∖ {0}) ∧ -1 ∈ (ℂ ∖ {0})) → {1, -1} ⊆ (ℂ ∖ {0})) | |
15 | 9, 13, 14 | mp2an 690 | . . . . 5 ⊢ {1, -1} ⊆ (ℂ ∖ {0}) |
16 | ressabs 16565 | . . . . 5 ⊢ (((ℂ ∖ {0}) ∈ V ∧ {1, -1} ⊆ (ℂ ∖ {0})) → (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1})) | |
17 | 5, 15, 16 | mp2an 690 | . . . 4 ⊢ (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) = ((mulGrp‘ℂfld) ↾s {1, -1}) |
18 | 3, 17 | eqtr4i 2849 | . . 3 ⊢ 𝑈 = (((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) ↾s {1, -1}) |
19 | 18 | subggrp 18284 | . 2 ⊢ ({1, -1} ∈ (SubGrp‘((mulGrp‘ℂfld) ↾s (ℂ ∖ {0}))) → 𝑈 ∈ Grp) |
20 | 2, 19 | ax-mp 5 | 1 ⊢ 𝑈 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 {cpr 4571 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 1c1 10540 -cneg 10873 ↾s cress 16486 Grpcgrp 18105 SubGrpcsubg 18275 mulGrpcmgp 19241 ℂfldccnfld 20547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-subg 18278 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-cnfld 20548 |
This theorem is referenced by: psgnghm 20726 evpmsubg 30791 |
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