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Mirrors > Home > ILE Home > Th. List > addcan | Unicode version |
Description: Cancellation law for addition. Theorem I.1 of [Apostol] p. 18. (Contributed by NM, 22-Nov-1994.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addcan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnegex2 7909 | . . 3 | |
2 | 1 | 3ad2ant1 987 | . 2 |
3 | oveq2 5750 | . . . 4 | |
4 | simprr 506 | . . . . . . 7 | |
5 | 4 | oveq1d 5757 | . . . . . 6 |
6 | simprl 505 | . . . . . . 7 | |
7 | simpl1 969 | . . . . . . 7 | |
8 | simpl2 970 | . . . . . . 7 | |
9 | 6, 7, 8 | addassd 7756 | . . . . . 6 |
10 | addid2 7869 | . . . . . . 7 | |
11 | 8, 10 | syl 14 | . . . . . 6 |
12 | 5, 9, 11 | 3eqtr3d 2158 | . . . . 5 |
13 | 4 | oveq1d 5757 | . . . . . 6 |
14 | simpl3 971 | . . . . . . 7 | |
15 | 6, 7, 14 | addassd 7756 | . . . . . 6 |
16 | addid2 7869 | . . . . . . 7 | |
17 | 14, 16 | syl 14 | . . . . . 6 |
18 | 13, 15, 17 | 3eqtr3d 2158 | . . . . 5 |
19 | 12, 18 | eqeq12d 2132 | . . . 4 |
20 | 3, 19 | syl5ib 153 | . . 3 |
21 | oveq2 5750 | . . 3 | |
22 | 20, 21 | impbid1 141 | . 2 |
23 | 2, 22 | rexlimddv 2531 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 wrex 2394 (class class class)co 5742 cc 7586 cc0 7588 caddc 7591 |
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-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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 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-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 |
This theorem is referenced by: addcani 7912 addcand 7914 subcan 7985 |
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