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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 7085 | . . . 4 | |
2 | 1 | 3ad2ant3 961 | . . 3 |
3 | oveq2 5540 | . . . . . . 7 | |
4 | 3 | adantl 271 | . . . . . 6 |
5 | simprl 497 | . . . . . . . . . 10 | |
6 | 5 | recnd 7147 | . . . . . . . . 9 |
7 | simpl3 943 | . . . . . . . . . 10 | |
8 | 7 | recnd 7147 | . . . . . . . . 9 |
9 | simpl1 941 | . . . . . . . . . 10 | |
10 | 9 | recnd 7147 | . . . . . . . . 9 |
11 | 6, 8, 10 | addassd 7141 | . . . . . . . 8 |
12 | simpl2 942 | . . . . . . . . . 10 | |
13 | 12 | recnd 7147 | . . . . . . . . 9 |
14 | 6, 8, 13 | addassd 7141 | . . . . . . . 8 |
15 | 11, 14 | eqeq12d 2095 | . . . . . . 7 |
16 | 15 | adantr 270 | . . . . . 6 |
17 | 4, 16 | mpbird 165 | . . . . 5 |
18 | 8 | adantr 270 | . . . . . . . . 9 |
19 | 6 | adantr 270 | . . . . . . . . 9 |
20 | addcom 7245 | . . . . . . . . 9 | |
21 | 18, 19, 20 | syl2anc 403 | . . . . . . . 8 |
22 | simplrr 502 | . . . . . . . 8 | |
23 | 21, 22 | eqtr3d 2115 | . . . . . . 7 |
24 | 23 | oveq1d 5547 | . . . . . 6 |
25 | 10 | adantr 270 | . . . . . . 7 |
26 | addid2 7247 | . . . . . . 7 | |
27 | 25, 26 | syl 14 | . . . . . 6 |
28 | 24, 27 | eqtrd 2113 | . . . . 5 |
29 | 23 | oveq1d 5547 | . . . . . 6 |
30 | 13 | adantr 270 | . . . . . . 7 |
31 | addid2 7247 | . . . . . . 7 | |
32 | 30, 31 | syl 14 | . . . . . 6 |
33 | 29, 32 | eqtrd 2113 | . . . . 5 |
34 | 17, 28, 33 | 3eqtr3d 2121 | . . . 4 |
35 | 34 | ex 113 | . . 3 |
36 | 2, 35 | rexlimddv 2481 | . 2 |
37 | oveq2 5540 | . 2 | |
38 | 36, 37 | impbid1 140 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wrex 2349 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 caddc 6984 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: (None) |
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