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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 7722 | . . . 4 | |
2 | 1 | 3ad2ant3 1004 | . . 3 |
3 | oveq2 5775 | . . . . . . 7 | |
4 | 3 | adantl 275 | . . . . . 6 |
5 | simprl 520 | . . . . . . . . . 10 | |
6 | 5 | recnd 7787 | . . . . . . . . 9 |
7 | simpl3 986 | . . . . . . . . . 10 | |
8 | 7 | recnd 7787 | . . . . . . . . 9 |
9 | simpl1 984 | . . . . . . . . . 10 | |
10 | 9 | recnd 7787 | . . . . . . . . 9 |
11 | 6, 8, 10 | addassd 7781 | . . . . . . . 8 |
12 | simpl2 985 | . . . . . . . . . 10 | |
13 | 12 | recnd 7787 | . . . . . . . . 9 |
14 | 6, 8, 13 | addassd 7781 | . . . . . . . 8 |
15 | 11, 14 | eqeq12d 2152 | . . . . . . 7 |
16 | 15 | adantr 274 | . . . . . 6 |
17 | 4, 16 | mpbird 166 | . . . . 5 |
18 | 8 | adantr 274 | . . . . . . . . 9 |
19 | 6 | adantr 274 | . . . . . . . . 9 |
20 | addcom 7892 | . . . . . . . . 9 | |
21 | 18, 19, 20 | syl2anc 408 | . . . . . . . 8 |
22 | simplrr 525 | . . . . . . . 8 | |
23 | 21, 22 | eqtr3d 2172 | . . . . . . 7 |
24 | 23 | oveq1d 5782 | . . . . . 6 |
25 | 10 | adantr 274 | . . . . . . 7 |
26 | addid2 7894 | . . . . . . 7 | |
27 | 25, 26 | syl 14 | . . . . . 6 |
28 | 24, 27 | eqtrd 2170 | . . . . 5 |
29 | 23 | oveq1d 5782 | . . . . . 6 |
30 | 13 | adantr 274 | . . . . . . 7 |
31 | addid2 7894 | . . . . . . 7 | |
32 | 30, 31 | syl 14 | . . . . . 6 |
33 | 29, 32 | eqtrd 2170 | . . . . 5 |
34 | 17, 28, 33 | 3eqtr3d 2178 | . . . 4 |
35 | 34 | ex 114 | . . 3 |
36 | 2, 35 | rexlimddv 2552 | . 2 |
37 | oveq2 5775 | . 2 | |
38 | 36, 37 | impbid1 141 | 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 cr 7612 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-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 |
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: (None) |
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