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Mirrors > Home > MPE Home > Th. List > idnghm | Structured version Visualization version GIF version |
Description: The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
idnghm.2 | ⊢ 𝑉 = (Base‘𝑆) |
Ref | Expression |
---|---|
idnghm | ⊢ (𝑆 ∈ NrmGrp → ( I ↾ 𝑉) ∈ (𝑆 NGHom 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . 5 ⊢ (𝑆 normOp 𝑆) = (𝑆 normOp 𝑆) | |
2 | idnghm.2 | . . . . 5 ⊢ 𝑉 = (Base‘𝑆) | |
3 | eqid 2738 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
4 | 1, 2, 3 | nmoid 23497 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} ⊊ 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) = 1) |
5 | 1re 10721 | . . . 4 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | eqeltrdi 2841 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} ⊊ 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ) |
7 | eleq2 2821 | . . . . . . . . . 10 ⊢ ({(0g‘𝑆)} = 𝑉 → (𝑥 ∈ {(0g‘𝑆)} ↔ 𝑥 ∈ 𝑉)) | |
8 | 7 | biimpar 481 | . . . . . . . . 9 ⊢ (({(0g‘𝑆)} = 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ {(0g‘𝑆)}) |
9 | elsni 4533 | . . . . . . . . 9 ⊢ (𝑥 ∈ {(0g‘𝑆)} → 𝑥 = (0g‘𝑆)) | |
10 | 8, 9 | syl 17 | . . . . . . . 8 ⊢ (({(0g‘𝑆)} = 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑥 = (0g‘𝑆)) |
11 | 10 | mpteq2dva 5125 | . . . . . . 7 ⊢ ({(0g‘𝑆)} = 𝑉 → (𝑥 ∈ 𝑉 ↦ 𝑥) = (𝑥 ∈ 𝑉 ↦ (0g‘𝑆))) |
12 | mptresid 5892 | . . . . . . 7 ⊢ ( I ↾ 𝑉) = (𝑥 ∈ 𝑉 ↦ 𝑥) | |
13 | fconstmpt 5585 | . . . . . . 7 ⊢ (𝑉 × {(0g‘𝑆)}) = (𝑥 ∈ 𝑉 ↦ (0g‘𝑆)) | |
14 | 11, 12, 13 | 3eqtr4g 2798 | . . . . . 6 ⊢ ({(0g‘𝑆)} = 𝑉 → ( I ↾ 𝑉) = (𝑉 × {(0g‘𝑆)})) |
15 | 14 | fveq2d 6680 | . . . . 5 ⊢ ({(0g‘𝑆)} = 𝑉 → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) = ((𝑆 normOp 𝑆)‘(𝑉 × {(0g‘𝑆)}))) |
16 | 1, 2, 3 | nmo0 23490 | . . . . . 6 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑆 ∈ NrmGrp) → ((𝑆 normOp 𝑆)‘(𝑉 × {(0g‘𝑆)})) = 0) |
17 | 16 | anidms 570 | . . . . 5 ⊢ (𝑆 ∈ NrmGrp → ((𝑆 normOp 𝑆)‘(𝑉 × {(0g‘𝑆)})) = 0) |
18 | 15, 17 | sylan9eqr 2795 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} = 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) = 0) |
19 | 0re 10723 | . . . 4 ⊢ 0 ∈ ℝ | |
20 | 18, 19 | eqeltrdi 2841 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} = 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ) |
21 | ngpgrp 23354 | . . . . . 6 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) | |
22 | 2, 3 | grpidcl 18251 | . . . . . 6 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝑉) |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ NrmGrp → (0g‘𝑆) ∈ 𝑉) |
24 | 23 | snssd 4697 | . . . 4 ⊢ (𝑆 ∈ NrmGrp → {(0g‘𝑆)} ⊆ 𝑉) |
25 | sspss 3990 | . . . 4 ⊢ ({(0g‘𝑆)} ⊆ 𝑉 ↔ ({(0g‘𝑆)} ⊊ 𝑉 ∨ {(0g‘𝑆)} = 𝑉)) | |
26 | 24, 25 | sylib 221 | . . 3 ⊢ (𝑆 ∈ NrmGrp → ({(0g‘𝑆)} ⊊ 𝑉 ∨ {(0g‘𝑆)} = 𝑉)) |
27 | 6, 20, 26 | mpjaodan 958 | . 2 ⊢ (𝑆 ∈ NrmGrp → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ) |
28 | id 22 | . . 3 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ NrmGrp) | |
29 | 2 | idghm 18493 | . . . 4 ⊢ (𝑆 ∈ Grp → ( I ↾ 𝑉) ∈ (𝑆 GrpHom 𝑆)) |
30 | 21, 29 | syl 17 | . . 3 ⊢ (𝑆 ∈ NrmGrp → ( I ↾ 𝑉) ∈ (𝑆 GrpHom 𝑆)) |
31 | 1 | isnghm2 23479 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑆 ∈ NrmGrp ∧ ( I ↾ 𝑉) ∈ (𝑆 GrpHom 𝑆)) → (( I ↾ 𝑉) ∈ (𝑆 NGHom 𝑆) ↔ ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ)) |
32 | 28, 30, 31 | mpd3an23 1464 | . 2 ⊢ (𝑆 ∈ NrmGrp → (( I ↾ 𝑉) ∈ (𝑆 NGHom 𝑆) ↔ ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ)) |
33 | 27, 32 | mpbird 260 | 1 ⊢ (𝑆 ∈ NrmGrp → ( I ↾ 𝑉) ∈ (𝑆 NGHom 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 ⊊ wpss 3844 {csn 4516 ↦ cmpt 5110 I cid 5428 × cxp 5523 ↾ cres 5527 ‘cfv 6339 (class class class)co 7172 ℝcr 10616 0cc0 10617 1c1 10618 Basecbs 16588 0gc0g 16818 Grpcgrp 18221 GrpHom cghm 18475 NrmGrpcngp 23332 normOp cnmo 23460 NGHom cnghm 23461 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
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 2075 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 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-sup 8981 df-inf 8982 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-n0 11979 df-z 12065 df-uz 12327 df-q 12433 df-rp 12475 df-xneg 12592 df-xadd 12593 df-xmul 12594 df-ico 12829 df-0g 16820 df-topgen 16822 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-mhm 18074 df-grp 18224 df-ghm 18476 df-psmet 20211 df-xmet 20212 df-met 20213 df-bl 20214 df-mopn 20215 df-top 21647 df-topon 21664 df-topsp 21686 df-bases 21699 df-xms 23075 df-ms 23076 df-nm 23337 df-ngp 23338 df-nmo 23463 df-nghm 23464 |
This theorem is referenced by: idnmhm 23509 |
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