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Mirrors > Home > HSE Home > Th. List > hvmulcomi | Structured version Visualization version GIF version |
Description: Scalar multiplication commutative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvmulcom.1 | โข ๐ด โ โ |
hvmulcom.2 | โข ๐ต โ โ |
hvmulcom.3 | โข ๐ถ โ โ |
Ref | Expression |
---|---|
hvmulcomi | โข (๐ด ยทโ (๐ต ยทโ ๐ถ)) = (๐ต ยทโ (๐ด ยทโ ๐ถ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvmulcom.1 | . 2 โข ๐ด โ โ | |
2 | hvmulcom.2 | . 2 โข ๐ต โ โ | |
3 | hvmulcom.3 | . 2 โข ๐ถ โ โ | |
4 | hvmulcom 30034 | . 2 โข ((๐ด โ โ โง ๐ต โ โ โง ๐ถ โ โ) โ (๐ด ยทโ (๐ต ยทโ ๐ถ)) = (๐ต ยทโ (๐ด ยทโ ๐ถ))) | |
5 | 1, 2, 3, 4 | mp3an 1462 | 1 โข (๐ด ยทโ (๐ต ยทโ ๐ถ)) = (๐ต ยทโ (๐ด ยทโ ๐ถ)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 โ wcel 2107 (class class class)co 7361 โcc 11057 โchba 29910 ยทโ csm 29912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-mulcom 11123 ax-hvmulass 29998 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: (None) |
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