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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 2823 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | subgss 18282 | . . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺)) |
3 | 2 | resmptd 5910 | . . 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 18285 | . . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 = (Base‘𝐻)) |
6 | eqid 2823 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
7 | 4, 6 | ressds 16688 | . . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (dist‘𝐺) = (dist‘𝐻)) |
8 | eqidd 2824 | . . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝑥 = 𝑥) | |
9 | eqid 2823 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
10 | 4, 9 | subg0 18287 | . . . . 5 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (0g‘𝐺) = (0g‘𝐻)) |
11 | 7, 8, 10 | oveq123d 7179 | . . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥(dist‘𝐺)(0g‘𝐺)) = (𝑥(dist‘𝐻)(0g‘𝐻))) |
12 | 5, 11 | mpteq12dv 5153 | . . 3 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (𝑥 ∈ 𝐴 ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) = (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻)))) |
13 | 3, 12 | eqtr2d 2859 | . 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 2823 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
16 | eqid 2823 | . . 3 ⊢ (0g‘𝐻) = (0g‘𝐻) | |
17 | eqid 2823 | . . 3 ⊢ (dist‘𝐻) = (dist‘𝐻) | |
18 | 14, 15, 16, 17 | nmfval 23200 | . 2 ⊢ 𝑀 = (𝑥 ∈ (Base‘𝐻) ↦ (𝑥(dist‘𝐻)(0g‘𝐻))) |
19 | subgnm.n | . . . 4 ⊢ 𝑁 = (norm‘𝐺) | |
20 | 19, 1, 9, 6 | nmfval 23200 | . . 3 ⊢ 𝑁 = (𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) |
21 | 20 | reseq1i 5851 | . 2 ⊢ (𝑁 ↾ 𝐴) = ((𝑥 ∈ (Base‘𝐺) ↦ (𝑥(dist‘𝐺)(0g‘𝐺))) ↾ 𝐴) |
22 | 13, 18, 21 | 3eqtr4g 2883 | 1 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝑀 = (𝑁 ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5148 ↾ cres 5559 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 ↾s cress 16486 distcds 16576 0gc0g 16715 SubGrpcsubg 18275 normcnm 23188 |
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-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 |
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-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-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 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-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-ds 16589 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-subg 18278 df-nm 23194 |
This theorem is referenced by: subgnm2 23245 subrgnrg 23284 isncvsngp 23755 cphsscph 23856 |
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