Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > addsubass | Unicode version |
Description: Associative-type law for addition and subtraction. (Contributed by NM, 6-Aug-2003.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addsubass |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 981 | . . . . 5 | |
2 | subcl 7961 | . . . . . 6 | |
3 | 2 | 3adant1 999 | . . . . 5 |
4 | simp3 983 | . . . . 5 | |
5 | 1, 3, 4 | addassd 7788 | . . . 4 |
6 | npcan 7971 | . . . . . 6 | |
7 | 6 | 3adant1 999 | . . . . 5 |
8 | 7 | oveq2d 5790 | . . . 4 |
9 | 5, 8 | eqtrd 2172 | . . 3 |
10 | 9 | oveq1d 5789 | . 2 |
11 | 1, 3 | addcld 7785 | . . 3 |
12 | pncan 7968 | . . 3 | |
13 | 11, 4, 12 | syl2anc 408 | . 2 |
14 | 10, 13 | eqtr3d 2174 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wceq 1331 wcel 1480 (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: addsub 7973 subadd23 7974 addsubeq4 7977 npncan 7983 subsub 7992 subsub3 7994 addsub4 8005 negsub 8010 addsubassi 8053 addsubassd 8093 zeo 9156 frecfzen2 10200 odd2np1 11570 |
Copyright terms: Public domain | W3C validator |