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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 7356 | . . 3 | |
2 | negsub 7356 | . . 3 | |
3 | 1, 2 | oveqan12d 5551 | . 2 |
4 | negcl 7308 | . . . 4 | |
5 | negcl 7308 | . . . . 5 | |
6 | muladd 7488 | . . . . 5 | |
7 | 5, 6 | sylanr2 397 | . . . 4 |
8 | 4, 7 | sylanl2 395 | . . 3 |
9 | mul2neg 7502 | . . . . . . 7 | |
10 | 9 | ancoms 264 | . . . . . 6 |
11 | 10 | oveq2d 5548 | . . . . 5 |
12 | 11 | ad2ant2l 491 | . . . 4 |
13 | mulneg2 7500 | . . . . . . . 8 | |
14 | mulneg2 7500 | . . . . . . . 8 | |
15 | 13, 14 | oveqan12d 5551 | . . . . . . 7 |
16 | mulcl 7100 | . . . . . . . 8 | |
17 | mulcl 7100 | . . . . . . . 8 | |
18 | negdi 7365 | . . . . . . . 8 | |
19 | 16, 17, 18 | syl2an 283 | . . . . . . 7 |
20 | 15, 19 | eqtr4d 2116 | . . . . . 6 |
21 | 20 | ancom2s 530 | . . . . 5 |
22 | 21 | an42s 553 | . . . 4 |
23 | 12, 22 | oveq12d 5550 | . . 3 |
24 | mulcl 7100 | . . . . . 6 | |
25 | mulcl 7100 | . . . . . . 7 | |
26 | 25 | ancoms 264 | . . . . . 6 |
27 | addcl 7098 | . . . . . 6 | |
28 | 24, 26, 27 | syl2an 283 | . . . . 5 |
29 | 28 | an4s 552 | . . . 4 |
30 | 17 | ancoms 264 | . . . . . 6 |
31 | addcl 7098 | . . . . . 6 | |
32 | 16, 30, 31 | syl2an 283 | . . . . 5 |
33 | 32 | an42s 553 | . . . 4 |
34 | 29, 33 | negsubd 7425 | . . 3 |
35 | 8, 23, 34 | 3eqtrd 2117 | . 2 |
36 | 3, 35 | eqtr3d 2115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 caddc 6984 cmul 6986 cmin 7279 cneg 7280 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 |
This theorem is referenced by: mulsubd 7521 muleqadd 7758 addltmul 8267 sqabssub 9942 |
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