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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 986 | . . . . 5 | |
2 | simp3 988 | . . . . 5 | |
3 | subcl 8088 | . . . . . 6 | |
4 | 3 | 3adant1 1004 | . . . . 5 |
5 | 1, 2, 4 | adddid 7914 | . . . 4 |
6 | pncan3 8097 | . . . . . . 7 | |
7 | 6 | ancoms 266 | . . . . . 6 |
8 | 7 | 3adant1 1004 | . . . . 5 |
9 | 8 | oveq2d 5852 | . . . 4 |
10 | 5, 9 | eqtr3d 2199 | . . 3 |
11 | mulcl 7871 | . . . . 5 | |
12 | 11 | 3adant3 1006 | . . . 4 |
13 | mulcl 7871 | . . . . 5 | |
14 | 13 | 3adant2 1005 | . . . 4 |
15 | mulcl 7871 | . . . . . 6 | |
16 | 3, 15 | sylan2 284 | . . . . 5 |
17 | 16 | 3impb 1188 | . . . 4 |
18 | 12, 14, 17 | subaddd 8218 | . . 3 |
19 | 10, 18 | mpbird 166 | . 2 |
20 | 19 | eqcomd 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 967 wceq 1342 wcel 2135 (class class class)co 5836 cc 7742 caddc 7747 cmul 7749 cmin 8060 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 |
This theorem is referenced by: subdir 8275 subdii 8296 subdid 8303 expubnd 10502 subsq 10551 cos01bnd 11685 modmulconst 11749 odd2np1 11795 omoe 11818 omeo 11820 phiprmpw 12131 pythagtriplem14 12186 |
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