Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > subdi | Unicode version |
Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.) |
Ref | Expression |
---|---|
subdi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . . . . 5 | |
2 | simp3 983 | . . . . 5 | |
3 | subcl 7961 | . . . . . 6 | |
4 | 3 | 3adant1 999 | . . . . 5 |
5 | 1, 2, 4 | adddid 7790 | . . . 4 |
6 | pncan3 7970 | . . . . . . 7 | |
7 | 6 | ancoms 266 | . . . . . 6 |
8 | 7 | 3adant1 999 | . . . . 5 |
9 | 8 | oveq2d 5790 | . . . 4 |
10 | 5, 9 | eqtr3d 2174 | . . 3 |
11 | mulcl 7747 | . . . . 5 | |
12 | 11 | 3adant3 1001 | . . . 4 |
13 | mulcl 7747 | . . . . 5 | |
14 | 13 | 3adant2 1000 | . . . 4 |
15 | mulcl 7747 | . . . . . 6 | |
16 | 3, 15 | sylan2 284 | . . . . 5 |
17 | 16 | 3impb 1177 | . . . 4 |
18 | 12, 14, 17 | subaddd 8091 | . . 3 |
19 | 10, 18 | mpbird 166 | . 2 |
20 | 19 | eqcomd 2145 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 caddc 7623 cmul 7625 cmin 7933 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sub 7935 |
This theorem is referenced by: subdir 8148 subdii 8169 subdid 8176 expubnd 10350 subsq 10399 cos01bnd 11465 modmulconst 11525 odd2np1 11570 omoe 11593 omeo 11595 phiprmpw 11898 |
Copyright terms: Public domain | W3C validator |