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Mirrors > Home > MPE Home > Th. List > mul32i | Structured version Visualization version Unicode version |
Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.) |
Ref | Expression |
---|---|
mul.1 | |
mul.2 | |
mul.3 |
Ref | Expression |
---|---|
mul32i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul.1 | . 2 | |
2 | mul.2 | . 2 | |
3 | mul.3 | . 2 | |
4 | mul32 10203 | . 2 | |
5 | 1, 2, 3, 4 | mp3an 1424 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 (class class class)co 6650 cc 9934 cmul 9941 |
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-mulcom 10000 ax-mulass 10002 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 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 |
This theorem is referenced by: 8th4div3 11252 faclbnd4lem1 13080 bpoly4 14790 dec5nprm 15770 dec2nprm 15771 karatsuba 15792 karatsubaOLD 15793 quart1lem 24582 log2ublem2 24674 log2ub 24676 normlem3 27969 bcseqi 27977 dpmul100 29605 dpmul1000 29607 |
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