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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 6979 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇)) | |
2 | nmofval.1 | . . . . . 6 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
3 | 1, 2 | syl6eqr 2826 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁) |
4 | 3 | cnveqd 5589 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ◡(𝑠 normOp 𝑡) = ◡𝑁) |
5 | 4 | imaeq1d 5763 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (◡(𝑠 normOp 𝑡) “ ℝ) = (◡𝑁 “ ℝ)) |
6 | df-nghm 23011 | . . 3 ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) | |
7 | 2 | ovexi 7003 | . . . . 5 ⊢ 𝑁 ∈ V |
8 | 7 | cnvex 7439 | . . . 4 ⊢ ◡𝑁 ∈ V |
9 | 8 | imaex 7430 | . . 3 ⊢ (◡𝑁 “ ℝ) ∈ V |
10 | 5, 6, 9 | ovmpoa 7115 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
11 | 6 | mpondm0 7199 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅) |
12 | nmoffn 23013 | . . . . . . . . . 10 ⊢ normOp Fn (NrmGrp × NrmGrp) | |
13 | fndm 6282 | . . . . . . . . . 10 ⊢ ( normOp Fn (NrmGrp × NrmGrp) → dom normOp = (NrmGrp × NrmGrp)) | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . 9 ⊢ dom normOp = (NrmGrp × NrmGrp) |
15 | 14 | ndmov 7142 | . . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅) |
16 | 2, 15 | syl5eq 2820 | . . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅) |
17 | 16 | cnveqd 5589 | . . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ◡∅) |
18 | cnv0 5833 | . . . . . 6 ⊢ ◡∅ = ∅ | |
19 | 17, 18 | syl6eq 2824 | . . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ∅) |
20 | 19 | imaeq1d 5763 | . . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = (∅ “ ℝ)) |
21 | 0ima 5780 | . . . 4 ⊢ (∅ “ ℝ) = ∅ | |
22 | 20, 21 | syl6eq 2824 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = ∅) |
23 | 11, 22 | eqtr4d 2811 | . 2 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
24 | 10, 23 | pm2.61i 177 | 1 ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∅c0 4173 × cxp 5398 ◡ccnv 5399 dom cdm 5400 “ cima 5403 Fn wfn 6177 (class class class)co 6970 ℝcr 10326 NrmGrpcngp 22880 normOp cnmo 23007 NGHom cnghm 23008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-1st 7494 df-2nd 7495 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-ico 12553 df-nmo 23010 df-nghm 23011 |
This theorem is referenced by: isnghm 23025 |
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