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Mirrors > Home > ILE Home > Th. List > subadd | Unicode version |
Description: Relationship between subtraction and addition. (Contributed by NM, 20-Jan-1997.) (Revised by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
subadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subval 7922 | . . . 4 | |
2 | 1 | eqeq1d 2126 | . . 3 |
3 | 2 | 3adant3 986 | . 2 |
4 | negeu 7921 | . . . . 5 | |
5 | oveq2 5750 | . . . . . . 7 | |
6 | 5 | eqeq1d 2126 | . . . . . 6 |
7 | 6 | riota2 5720 | . . . . 5 |
8 | 4, 7 | sylan2 284 | . . . 4 |
9 | 8 | 3impb 1162 | . . 3 |
10 | 9 | 3com13 1171 | . 2 |
11 | 3, 10 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wreu 2395 crio 5697 (class class class)co 5742 cc 7586 caddc 7591 cmin 7901 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 |
This theorem is referenced by: subadd2 7934 subsub23 7935 pncan 7936 pncan3 7938 addsubeq4 7945 subsub2 7958 renegcl 7991 subaddi 8017 subaddd 8059 fzen 9791 nn0ennn 10174 cos2t 11384 cos2tsin 11385 odd2np1 11497 divalgb 11549 sincosq1eq 12847 coskpi 12856 |
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