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Mirrors > Home > MPE Home > Th. List > nmsub | Structured version Visualization version GIF version |
Description: The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nmf.x | ⊢ 𝑋 = (Base‘𝐺) |
nmf.n | ⊢ 𝑁 = (norm‘𝐺) |
nmmtri.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
nmsub | ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − 𝐵)) = (𝑁‘(𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1137 | . . . . 5 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ NrmGrp) | |
2 | ngpgrp 23353 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐺 ∈ Grp) |
4 | simp3 1139 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
5 | simp2 1138 | . . . 4 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
6 | nmf.x | . . . . 5 ⊢ 𝑋 = (Base‘𝐺) | |
7 | nmmtri.m | . . . . 5 ⊢ − = (-g‘𝐺) | |
8 | eqid 2738 | . . . . 5 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
9 | 6, 7, 8 | grpinvsub 18300 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 − 𝐴)) = (𝐴 − 𝐵)) |
10 | 3, 4, 5, 9 | syl3anc 1372 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((invg‘𝐺)‘(𝐵 − 𝐴)) = (𝐴 − 𝐵)) |
11 | 10 | fveq2d 6679 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘((invg‘𝐺)‘(𝐵 − 𝐴))) = (𝑁‘(𝐴 − 𝐵))) |
12 | 6, 7 | grpsubcl 18298 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 − 𝐴) ∈ 𝑋) |
13 | 3, 4, 5, 12 | syl3anc 1372 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵 − 𝐴) ∈ 𝑋) |
14 | nmf.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
15 | 6, 14, 8 | nminv 23375 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐵 − 𝐴) ∈ 𝑋) → (𝑁‘((invg‘𝐺)‘(𝐵 − 𝐴))) = (𝑁‘(𝐵 − 𝐴))) |
16 | 1, 13, 15 | syl2anc 587 | . 2 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘((invg‘𝐺)‘(𝐵 − 𝐴))) = (𝑁‘(𝐵 − 𝐴))) |
17 | 11, 16 | eqtr3d 2775 | 1 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − 𝐵)) = (𝑁‘(𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2113 ‘cfv 6340 (class class class)co 7171 Basecbs 16587 Grpcgrp 18220 invgcminusg 18221 -gcsg 18222 normcnm 23330 NrmGrpcngp 23331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7480 ax-cnex 10672 ax-resscn 10673 ax-1cn 10674 ax-icn 10675 ax-addcl 10676 ax-addrcl 10677 ax-mulcl 10678 ax-mulrcl 10679 ax-mulcom 10680 ax-addass 10681 ax-mulass 10682 ax-distr 10683 ax-i2m1 10684 ax-1ne0 10685 ax-1rid 10686 ax-rnegex 10687 ax-rrecex 10688 ax-cnre 10689 ax-pre-lttri 10690 ax-pre-lttrn 10691 ax-pre-ltadd 10692 ax-pre-mulgt0 10693 ax-pre-sup 10694 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3683 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7128 df-ov 7174 df-oprab 7175 df-mpo 7176 df-om 7601 df-1st 7715 df-2nd 7716 df-wrecs 7977 df-recs 8038 df-rdg 8076 df-er 8321 df-map 8440 df-en 8557 df-dom 8558 df-sdom 8559 df-sup 8980 df-inf 8981 df-pnf 10756 df-mnf 10757 df-xr 10758 df-ltxr 10759 df-le 10760 df-sub 10951 df-neg 10952 df-div 11377 df-nn 11718 df-2 11780 df-n0 11978 df-z 12064 df-uz 12326 df-q 12432 df-rp 12474 df-xneg 12591 df-xadd 12592 df-xmul 12593 df-0g 16819 df-topgen 16821 df-mgm 17969 df-sgrp 18018 df-mnd 18029 df-grp 18223 df-minusg 18224 df-sbg 18225 df-psmet 20210 df-xmet 20211 df-met 20212 df-bl 20213 df-mopn 20214 df-top 21646 df-topon 21663 df-topsp 21685 df-bases 21698 df-xms 23074 df-ms 23075 df-nm 23336 df-ngp 23337 |
This theorem is referenced by: ncvsdif 23908 |
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