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Mirrors > Home > MPE Home > Th. List > nghmfval | Structured version Visualization version GIF version |
Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmofval.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
Ref | Expression |
---|---|
nghmfval | ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7159 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇)) | |
2 | nmofval.1 | . . . . . 6 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
3 | 1, 2 | syl6eqr 2874 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁) |
4 | 3 | cnveqd 5740 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ◡(𝑠 normOp 𝑡) = ◡𝑁) |
5 | 4 | imaeq1d 5922 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (◡(𝑠 normOp 𝑡) “ ℝ) = (◡𝑁 “ ℝ)) |
6 | df-nghm 23312 | . . 3 ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) | |
7 | 2 | ovexi 7184 | . . . . 5 ⊢ 𝑁 ∈ V |
8 | 7 | cnvex 7624 | . . . 4 ⊢ ◡𝑁 ∈ V |
9 | 8 | imaex 7615 | . . 3 ⊢ (◡𝑁 “ ℝ) ∈ V |
10 | 5, 6, 9 | ovmpoa 7299 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
11 | 6 | mpondm0 7380 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅) |
12 | nmoffn 23314 | . . . . . . . . . 10 ⊢ normOp Fn (NrmGrp × NrmGrp) | |
13 | fndm 6449 | . . . . . . . . . 10 ⊢ ( normOp Fn (NrmGrp × NrmGrp) → dom normOp = (NrmGrp × NrmGrp)) | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . 9 ⊢ dom normOp = (NrmGrp × NrmGrp) |
15 | 14 | ndmov 7326 | . . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅) |
16 | 2, 15 | syl5eq 2868 | . . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅) |
17 | 16 | cnveqd 5740 | . . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ◡∅) |
18 | cnv0 5993 | . . . . . 6 ⊢ ◡∅ = ∅ | |
19 | 17, 18 | syl6eq 2872 | . . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ∅) |
20 | 19 | imaeq1d 5922 | . . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = (∅ “ ℝ)) |
21 | 0ima 5940 | . . . 4 ⊢ (∅ “ ℝ) = ∅ | |
22 | 20, 21 | syl6eq 2872 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = ∅) |
23 | 11, 22 | eqtr4d 2859 | . 2 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
24 | 10, 23 | pm2.61i 184 | 1 ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∅c0 4290 × cxp 5547 ◡ccnv 5548 dom cdm 5549 “ cima 5552 Fn wfn 6344 (class class class)co 7150 ℝcr 10530 NrmGrpcngp 23181 normOp cnmo 23308 NGHom cnghm 23309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-ico 12738 df-nmo 23311 df-nghm 23312 |
This theorem is referenced by: isnghm 23326 |
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