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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 2798 | . . . . 5 ⊢ (𝑆 normOp 𝑆) = (𝑆 normOp 𝑆) | |
2 | idnghm.2 | . . . . 5 ⊢ 𝑉 = (Base‘𝑆) | |
3 | eqid 2798 | . . . . 5 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
4 | 1, 2, 3 | nmoid 23348 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} ⊊ 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) = 1) |
5 | 1re 10630 | . . . 4 ⊢ 1 ∈ ℝ | |
6 | 4, 5 | eqeltrdi 2898 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} ⊊ 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ) |
7 | eleq2 2878 | . . . . . . . . . 10 ⊢ ({(0g‘𝑆)} = 𝑉 → (𝑥 ∈ {(0g‘𝑆)} ↔ 𝑥 ∈ 𝑉)) | |
8 | 7 | biimpar 481 | . . . . . . . . 9 ⊢ (({(0g‘𝑆)} = 𝑉 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ {(0g‘𝑆)}) |
9 | elsni 4542 | . . . . . . . . 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 5885 | . . . . . . 7 ⊢ ( I ↾ 𝑉) = (𝑥 ∈ 𝑉 ↦ 𝑥) | |
13 | fconstmpt 5578 | . . . . . . 7 ⊢ (𝑉 × {(0g‘𝑆)}) = (𝑥 ∈ 𝑉 ↦ (0g‘𝑆)) | |
14 | 11, 12, 13 | 3eqtr4g 2858 | . . . . . 6 ⊢ ({(0g‘𝑆)} = 𝑉 → ( I ↾ 𝑉) = (𝑉 × {(0g‘𝑆)})) |
15 | 14 | fveq2d 6649 | . . . . 5 ⊢ ({(0g‘𝑆)} = 𝑉 → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) = ((𝑆 normOp 𝑆)‘(𝑉 × {(0g‘𝑆)}))) |
16 | 1, 2, 3 | nmo0 23341 | . . . . . 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 2855 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} = 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) = 0) |
19 | 0re 10632 | . . . 4 ⊢ 0 ∈ ℝ | |
20 | 18, 19 | eqeltrdi 2898 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ {(0g‘𝑆)} = 𝑉) → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ) |
21 | ngpgrp 23205 | . . . . . 6 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ Grp) | |
22 | 2, 3 | grpidcl 18123 | . . . . . 6 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝑉) |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ (𝑆 ∈ NrmGrp → (0g‘𝑆) ∈ 𝑉) |
24 | 23 | snssd 4702 | . . . 4 ⊢ (𝑆 ∈ NrmGrp → {(0g‘𝑆)} ⊆ 𝑉) |
25 | sspss 4027 | . . . 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 956 | . 2 ⊢ (𝑆 ∈ NrmGrp → ((𝑆 normOp 𝑆)‘( I ↾ 𝑉)) ∈ ℝ) |
28 | id 22 | . . 3 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ NrmGrp) | |
29 | 2 | idghm 18365 | . . . 4 ⊢ (𝑆 ∈ Grp → ( I ↾ 𝑉) ∈ (𝑆 GrpHom 𝑆)) |
30 | 21, 29 | syl 17 | . . 3 ⊢ (𝑆 ∈ NrmGrp → ( I ↾ 𝑉) ∈ (𝑆 GrpHom 𝑆)) |
31 | 1 | isnghm2 23330 | . . 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 1460 | . 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 844 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ⊊ wpss 3882 {csn 4525 ↦ cmpt 5110 I cid 5424 × cxp 5517 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 Basecbs 16475 0gc0g 16705 Grpcgrp 18095 GrpHom cghm 18347 NrmGrpcngp 23184 normOp cnmo 23311 NGHom cnghm 23312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-0g 16707 df-topgen 16709 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-ghm 18348 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-xms 22927 df-ms 22928 df-nm 23189 df-ngp 23190 df-nmo 23314 df-nghm 23315 |
This theorem is referenced by: idnmhm 23360 |
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