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Mirrors > Home > ILE Home > Th. List > submul2 | Unicode version |
Description: Convert a subtraction to addition using multiplication by a negative. (Contributed by NM, 2-Feb-2007.) |
Ref | Expression |
---|---|
submul2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulneg2 8276 | . . . . 5 | |
2 | 1 | adantl 275 | . . . 4 |
3 | 2 | oveq2d 5843 | . . 3 |
4 | mulcl 7862 | . . . 4 | |
5 | negsub 8128 | . . . 4 | |
6 | 4, 5 | sylan2 284 | . . 3 |
7 | 3, 6 | eqtr2d 2191 | . 2 |
8 | 7 | 3impb 1181 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 (class class class)co 5827 cc 7733 caddc 7738 cmul 7740 cmin 8051 cneg 8052 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-setind 4499 ax-resscn 7827 ax-1cn 7828 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-sub 8053 df-neg 8054 |
This theorem is referenced by: cjap 10818 |
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