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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 7954 | . . . 4 | |
2 | 1 | eqeq1d 2148 | . . 3 |
3 | 2 | 3adant3 1001 | . 2 |
4 | negeu 7953 | . . . . 5 | |
5 | oveq2 5782 | . . . . . . 7 | |
6 | 5 | eqeq1d 2148 | . . . . . 6 |
7 | 6 | riota2 5752 | . . . . 5 |
8 | 4, 7 | sylan2 284 | . . . 4 |
9 | 8 | 3impb 1177 | . . 3 |
10 | 9 | 3com13 1186 | . 2 |
11 | 3, 10 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wreu 2418 crio 5729 (class class class)co 5774 cc 7618 caddc 7623 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: subadd2 7966 subsub23 7967 pncan 7968 pncan3 7970 addsubeq4 7977 subsub2 7990 renegcl 8023 subaddi 8049 subaddd 8091 fzen 9823 nn0ennn 10206 cos2t 11457 cos2tsin 11458 odd2np1 11570 divalgb 11622 sincosq1eq 12920 coskpi 12929 |
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