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Mirrors > Home > MPE Home > Th. List > nmmul | Structured version Visualization version GIF version |
Description: The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
nmmul.x | ⊢ 𝑋 = (Base‘𝑅) |
nmmul.n | ⊢ 𝑁 = (norm‘𝑅) |
nmmul.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
nmmul | ⊢ ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁‘𝐴) · (𝑁‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmmul.n | . . 3 ⊢ 𝑁 = (norm‘𝑅) | |
2 | eqid 2821 | . . 3 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅) | |
3 | 1, 2 | nrgabv 23269 | . 2 ⊢ (𝑅 ∈ NrmRing → 𝑁 ∈ (AbsVal‘𝑅)) |
4 | nmmul.x | . . 3 ⊢ 𝑋 = (Base‘𝑅) | |
5 | nmmul.t | . . 3 ⊢ · = (.r‘𝑅) | |
6 | 2, 4, 5 | abvmul 19599 | . 2 ⊢ ((𝑁 ∈ (AbsVal‘𝑅) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁‘𝐴) · (𝑁‘𝐵))) |
7 | 3, 6 | syl3an1 1159 | 1 ⊢ ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁‘𝐴) · (𝑁‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 · cmul 10541 Basecbs 16482 .rcmulr 16565 AbsValcabv 19586 normcnm 23185 NrmRingcnrg 23188 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-abv 19587 df-nrg 23194 |
This theorem is referenced by: nrgdsdi 23273 nrgdsdir 23274 nminvr 23277 nmdvr 23278 nrginvrcnlem 23299 |
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