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Mirrors > Home > ILE Home > Th. List > mulsub | Unicode version |
Description: Product of two differences. (Contributed by NM, 14-Jan-2006.) |
Ref | Expression |
---|---|
mulsub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negsub 8003 | . . 3 | |
2 | negsub 8003 | . . 3 | |
3 | 1, 2 | oveqan12d 5786 | . 2 |
4 | negcl 7955 | . . . 4 | |
5 | negcl 7955 | . . . . 5 | |
6 | muladd 8139 | . . . . 5 | |
7 | 5, 6 | sylanr2 402 | . . . 4 |
8 | 4, 7 | sylanl2 400 | . . 3 |
9 | mul2neg 8153 | . . . . . . 7 | |
10 | 9 | ancoms 266 | . . . . . 6 |
11 | 10 | oveq2d 5783 | . . . . 5 |
12 | 11 | ad2ant2l 499 | . . . 4 |
13 | mulneg2 8151 | . . . . . . . 8 | |
14 | mulneg2 8151 | . . . . . . . 8 | |
15 | 13, 14 | oveqan12d 5786 | . . . . . . 7 |
16 | mulcl 7740 | . . . . . . . 8 | |
17 | mulcl 7740 | . . . . . . . 8 | |
18 | negdi 8012 | . . . . . . . 8 | |
19 | 16, 17, 18 | syl2an 287 | . . . . . . 7 |
20 | 15, 19 | eqtr4d 2173 | . . . . . 6 |
21 | 20 | ancom2s 555 | . . . . 5 |
22 | 21 | an42s 578 | . . . 4 |
23 | 12, 22 | oveq12d 5785 | . . 3 |
24 | mulcl 7740 | . . . . . 6 | |
25 | mulcl 7740 | . . . . . . 7 | |
26 | 25 | ancoms 266 | . . . . . 6 |
27 | addcl 7738 | . . . . . 6 | |
28 | 24, 26, 27 | syl2an 287 | . . . . 5 |
29 | 28 | an4s 577 | . . . 4 |
30 | 17 | ancoms 266 | . . . . . 6 |
31 | addcl 7738 | . . . . . 6 | |
32 | 16, 30, 31 | syl2an 287 | . . . . 5 |
33 | 32 | an42s 578 | . . . 4 |
34 | 29, 33 | negsubd 8072 | . . 3 |
35 | 8, 23, 34 | 3eqtrd 2174 | . 2 |
36 | 3, 35 | eqtr3d 2172 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5767 cc 7611 caddc 7616 cmul 7618 cmin 7926 cneg 7927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-neg 7929 |
This theorem is referenced by: mulsubd 8172 muleqadd 8422 addltmul 8949 sqabssub 10821 |
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