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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdvsass | Structured version Visualization version Unicode version |
Description: Associative law for scalar product. (ax-hvmulass 27864 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 22-Sep-2015.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdvsass.v | |
slmdvsass.f | Scalar |
slmdvsass.s | |
slmdvsass.k | |
slmdvsass.t |
Ref | Expression |
---|---|
slmdvsass | SLMod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slmdvsass.v | . . . . . . . 8 | |
2 | eqid 2622 | . . . . . . . 8 | |
3 | slmdvsass.s | . . . . . . . 8 | |
4 | eqid 2622 | . . . . . . . 8 | |
5 | slmdvsass.f | . . . . . . . 8 Scalar | |
6 | slmdvsass.k | . . . . . . . 8 | |
7 | eqid 2622 | . . . . . . . 8 | |
8 | slmdvsass.t | . . . . . . . 8 | |
9 | eqid 2622 | . . . . . . . 8 | |
10 | eqid 2622 | . . . . . . . 8 | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | slmdlema 29756 | . . . . . . 7 SLMod |
12 | 11 | simprd 479 | . . . . . 6 SLMod |
13 | 12 | simp1d 1073 | . . . . 5 SLMod |
14 | 13 | 3expa 1265 | . . . 4 SLMod |
15 | 14 | anabsan2 863 | . . 3 SLMod |
16 | 15 | exp42 639 | . 2 SLMod |
17 | 16 | 3imp2 1282 | 1 SLMod |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cmulr 15942 Scalarcsca 15944 cvsca 15945 c0g 16100 cur 18501 SLModcslmd 29753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-slmd 29754 |
This theorem is referenced by: slmdvs0 29778 |
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