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Mirrors > Home > MPE Home > Th. List > nmval | Structured version Visualization version GIF version |
Description: The value of the norm function. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
nmfval.n | ⊢ 𝑁 = (norm‘𝑊) |
nmfval.x | ⊢ 𝑋 = (Base‘𝑊) |
nmfval.z | ⊢ 0 = (0g‘𝑊) |
nmfval.d | ⊢ 𝐷 = (dist‘𝑊) |
Ref | Expression |
---|---|
nmval | ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7165 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥𝐷 0 ) = (𝐴𝐷 0 )) | |
2 | nmfval.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
3 | nmfval.x | . . 3 ⊢ 𝑋 = (Base‘𝑊) | |
4 | nmfval.z | . . 3 ⊢ 0 = (0g‘𝑊) | |
5 | nmfval.d | . . 3 ⊢ 𝐷 = (dist‘𝑊) | |
6 | 2, 3, 4, 5 | nmfval 23200 | . 2 ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) |
7 | ovex 7191 | . 2 ⊢ (𝐴𝐷 0 ) ∈ V | |
8 | 1, 6, 7 | fvmpt 6770 | 1 ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 distcds 16576 0gc0g 16715 normcnm 23188 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 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-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 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-fv 6365 df-ov 7161 df-nm 23194 |
This theorem is referenced by: nmval2 23203 ngpds2 23217 isngp4 23223 nmge0 23228 nmeq0 23229 nminv 23232 nmmtri 23233 nmrtri 23235 |
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