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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 2728 | . . 3 β’ (AbsValβπ ) = (AbsValβπ ) | |
3 | 1, 2 | nrgabv 24571 | . 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 20702 | . 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 1161 | 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 1085 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 Β· cmul 11137 Basecbs 17173 .rcmulr 17227 AbsValcabv 20689 normcnm 24478 NrmRingcnrg 24481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-map 8840 df-abv 20690 df-nrg 24487 |
This theorem is referenced by: nrgdsdi 24575 nrgdsdir 24576 nminvr 24579 nmdvr 24580 nrginvrcnlem 24601 |
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