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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 | β’ π = (Baseβπ ) |
nmmul.n | β’ π = (normβπ ) |
nmmul.t | β’ Β· = (.rβπ ) |
Ref | Expression |
---|---|
nmmul | β’ ((π β NrmRing β§ π΄ β π β§ π΅ β π) β (πβ(π΄ Β· π΅)) = ((πβπ΄) Β· (πβπ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmmul.n | . . 3 β’ π = (normβπ ) | |
2 | eqid 2736 | . . 3 β’ (AbsValβπ ) = (AbsValβπ ) | |
3 | 1, 2 | nrgabv 23931 | . 2 β’ (π β NrmRing β π β (AbsValβπ )) |
4 | nmmul.x | . . 3 β’ π = (Baseβπ ) | |
5 | nmmul.t | . . 3 β’ Β· = (.rβπ ) | |
6 | 2, 4, 5 | abvmul 20195 | . 2 β’ ((π β (AbsValβπ ) β§ π΄ β π β§ π΅ β π) β (πβ(π΄ Β· π΅)) = ((πβπ΄) Β· (πβπ΅))) |
7 | 3, 6 | syl3an1 1162 | 1 β’ ((π β NrmRing β§ π΄ β π β§ π΅ β π) β (πβ(π΄ Β· π΅)) = ((πβπ΄) Β· (πβπ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6479 (class class class)co 7337 Β· cmul 10977 Basecbs 17009 .rcmulr 17060 AbsValcabv 20182 normcnm 23838 NrmRingcnrg 23841 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-map 8688 df-abv 20183 df-nrg 23847 |
This theorem is referenced by: nrgdsdi 23935 nrgdsdir 23936 nminvr 23939 nmdvr 23940 nrginvrcnlem 23961 |
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