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Mirrors > Home > ILE Home > Th. List > readdcan | Unicode version |
Description: Cancellation law for addition over the reals. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
readdcan |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rnegex 6993 | . . . 4 | |
2 | 1 | 3ad2ant3 927 | . . 3 |
3 | oveq2 5520 | . . . . . . 7 | |
4 | 3 | adantl 262 | . . . . . 6 |
5 | simprl 483 | . . . . . . . . . 10 | |
6 | 5 | recnd 7054 | . . . . . . . . 9 |
7 | simpl3 909 | . . . . . . . . . 10 | |
8 | 7 | recnd 7054 | . . . . . . . . 9 |
9 | simpl1 907 | . . . . . . . . . 10 | |
10 | 9 | recnd 7054 | . . . . . . . . 9 |
11 | 6, 8, 10 | addassd 7049 | . . . . . . . 8 |
12 | simpl2 908 | . . . . . . . . . 10 | |
13 | 12 | recnd 7054 | . . . . . . . . 9 |
14 | 6, 8, 13 | addassd 7049 | . . . . . . . 8 |
15 | 11, 14 | eqeq12d 2054 | . . . . . . 7 |
16 | 15 | adantr 261 | . . . . . 6 |
17 | 4, 16 | mpbird 156 | . . . . 5 |
18 | 8 | adantr 261 | . . . . . . . . 9 |
19 | 6 | adantr 261 | . . . . . . . . 9 |
20 | addcom 7150 | . . . . . . . . 9 | |
21 | 18, 19, 20 | syl2anc 391 | . . . . . . . 8 |
22 | simplrr 488 | . . . . . . . 8 | |
23 | 21, 22 | eqtr3d 2074 | . . . . . . 7 |
24 | 23 | oveq1d 5527 | . . . . . 6 |
25 | 10 | adantr 261 | . . . . . . 7 |
26 | addid2 7152 | . . . . . . 7 | |
27 | 25, 26 | syl 14 | . . . . . 6 |
28 | 24, 27 | eqtrd 2072 | . . . . 5 |
29 | 23 | oveq1d 5527 | . . . . . 6 |
30 | 13 | adantr 261 | . . . . . . 7 |
31 | addid2 7152 | . . . . . . 7 | |
32 | 30, 31 | syl 14 | . . . . . 6 |
33 | 29, 32 | eqtrd 2072 | . . . . 5 |
34 | 17, 28, 33 | 3eqtr3d 2080 | . . . 4 |
35 | 34 | ex 108 | . . 3 |
36 | 2, 35 | rexlimddv 2437 | . 2 |
37 | oveq2 5520 | . 2 | |
38 | 36, 37 | impbid1 130 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 wrex 2307 (class class class)co 5512 cc 6887 cr 6888 cc0 6889 caddc 6892 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-resscn 6976 ax-1cn 6977 ax-icn 6979 ax-addcl 6980 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: (None) |
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