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Mirrors > Home > MPE Home > Th. List > subgnm | Structured version Visualization version GIF version |
Description: The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
subgngp.h | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
subgnm.n | ⊢ 𝑁 = (norm‘𝐺) |
subgnm.m | ⊢ 𝑀 = (norm‘𝐻) |
Ref | Expression |
---|---|
subgnm | ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝑀 = (𝑁 ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | subgss 19082 | . . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺)) |
3 | 2 | resmptd 6044 | . . 3 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺)(0g‘𝐺)))) |
4 | subgngp.h | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
5 | 4 | subgbas 19085 | . . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 = (Base‘𝐻)) |
6 | eqid 2728 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
7 | 4, 6 | ressds 17391 | . . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (dist‘𝐺) = (dist‘𝐻)) |
8 | eqidd 2729 | . . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝑥 = 𝑥) | |
9 | eqid 2728 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
10 | 4, 9 | subg0 19087 | . . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻)) |
11 | 7, 8, 10 | oveq123d 7441 | . . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥(dist‘𝐺)(0g‘𝐺)) = (𝑥(dist‘𝐻)(0g‘𝐻))) |
12 | 5, 11 | mpteq12dv 5239 | . . 3 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) = (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻)))) |
13 | 3, 12 | eqtr2d 2769 | . 2 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻))) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) ↾ 𝐴)) |
14 | subgnm.m | . . 3 ⊢ 𝑀 = (norm‘𝐻) | |
15 | eqid 2728 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
16 | eqid 2728 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | eqid 2728 | . . 3 ⊢ (dist‘𝐻) = (dist‘𝐻) | |
18 | 14, 15, 16, 17 | nmfval 24510 | . 2 ⊢ 𝑀 = (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻))) |
19 | subgnm.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
20 | 19, 1, 9, 6 | nmfval 24510 | . . 3 ⊢ 𝑁 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) |
21 | 20 | reseq1i 5981 | . 2 ⊢ (𝑁 ↾ 𝐴) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) ↾ 𝐴) |
22 | 13, 18, 21 | 3eqtr4g 2793 | 1 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝑀 = (𝑁 ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ↦ cmpt 5231 ↾ cres 5680 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 ↾s cress 17209 distcds 17242 0gc0g 17421 SubGrpcsubg 19075 normcnm 24498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-ds 17255 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-subg 19078 df-nm 24504 |
This theorem is referenced by: subgnm2 24556 subrgnrg 24603 isncvsngp 25090 cphsscph 25192 |
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