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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 7167 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇)) | |
2 | nmofval.1 | . . . . . 6 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
3 | 1, 2 | syl6eqr 2876 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁) |
4 | 3 | cnveqd 5748 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ◡(𝑠 normOp 𝑡) = ◡𝑁) |
5 | 4 | imaeq1d 5930 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (◡(𝑠 normOp 𝑡) “ ℝ) = (◡𝑁 “ ℝ)) |
6 | df-nghm 23320 | . . 3 ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) | |
7 | 2 | ovexi 7192 | . . . . 5 ⊢ 𝑁 ∈ V |
8 | 7 | cnvex 7632 | . . . 4 ⊢ ◡𝑁 ∈ V |
9 | 8 | imaex 7623 | . . 3 ⊢ (◡𝑁 “ ℝ) ∈ V |
10 | 5, 6, 9 | ovmpoa 7307 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
11 | 6 | mpondm0 7388 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅) |
12 | nmoffn 23322 | . . . . . . . . . 10 ⊢ normOp Fn (NrmGrp × NrmGrp) | |
13 | fndm 6457 | . . . . . . . . . 10 ⊢ ( normOp Fn (NrmGrp × NrmGrp) → dom normOp = (NrmGrp × NrmGrp)) | |
14 | 12, 13 | ax-mp 5 | . . . . . . . . 9 ⊢ dom normOp = (NrmGrp × NrmGrp) |
15 | 14 | ndmov 7334 | . . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅) |
16 | 2, 15 | syl5eq 2870 | . . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅) |
17 | 16 | cnveqd 5748 | . . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ◡∅) |
18 | cnv0 6001 | . . . . . 6 ⊢ ◡∅ = ∅ | |
19 | 17, 18 | syl6eq 2874 | . . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ∅) |
20 | 19 | imaeq1d 5930 | . . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = (∅ “ ℝ)) |
21 | 0ima 5948 | . . . 4 ⊢ (∅ “ ℝ) = ∅ | |
22 | 20, 21 | syl6eq 2874 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = ∅) |
23 | 11, 22 | eqtr4d 2861 | . 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 1537 ∈ wcel 2114 ∅c0 4293 × cxp 5555 ◡ccnv 5556 dom cdm 5557 “ cima 5560 Fn wfn 6352 (class class class)co 7158 ℝcr 10538 NrmGrpcngp 23189 normOp cnmo 23316 NGHom cnghm 23317 |
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 ax-pre-sup 10617 |
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-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 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-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-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-ico 12747 df-nmo 23319 df-nghm 23320 |
This theorem is referenced by: isnghm 23334 |
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