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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 7934 | . . 3 | |
2 | 1 | 3ad2ant1 1002 | . 2 |
3 | oveq2 5775 | . . . 4 | |
4 | simprr 521 | . . . . . . 7 | |
5 | 4 | oveq1d 5782 | . . . . . 6 |
6 | simprl 520 | . . . . . . 7 | |
7 | simpl1 984 | . . . . . . 7 | |
8 | simpl2 985 | . . . . . . 7 | |
9 | 6, 7, 8 | addassd 7781 | . . . . . 6 |
10 | addid2 7894 | . . . . . . 7 | |
11 | 8, 10 | syl 14 | . . . . . 6 |
12 | 5, 9, 11 | 3eqtr3d 2178 | . . . . 5 |
13 | 4 | oveq1d 5782 | . . . . . 6 |
14 | simpl3 986 | . . . . . . 7 | |
15 | 6, 7, 14 | addassd 7781 | . . . . . 6 |
16 | addid2 7894 | . . . . . . 7 | |
17 | 14, 16 | syl 14 | . . . . . 6 |
18 | 13, 15, 17 | 3eqtr3d 2178 | . . . . 5 |
19 | 12, 18 | eqeq12d 2152 | . . . 4 |
20 | 3, 19 | syl5ib 153 | . . 3 |
21 | oveq2 5775 | . . 3 | |
22 | 20, 21 | impbid1 141 | . 2 |
23 | 2, 22 | rexlimddv 2552 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wrex 2415 (class class class)co 5767 cc 7611 cc0 7613 caddc 7616 |
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 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: addcani 7937 addcand 7939 subcan 8010 |
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