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Mirrors > Home > ILE Home > Th. List > add20 | Unicode version |
Description: Two nonnegative numbers are zero iff their sum is zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
add20 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 524 |
. . . . . . . . 9
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2 | simplrl 525 |
. . . . . . . . . 10
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3 | simplll 523 |
. . . . . . . . . 10
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4 | addge02 8259 |
. . . . . . . . . 10
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5 | 2, 3, 4 | syl2anc 409 |
. . . . . . . . 9
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6 | 1, 5 | mpbid 146 |
. . . . . . . 8
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7 | simpr 109 |
. . . . . . . 8
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8 | 6, 7 | breqtrd 3962 |
. . . . . . 7
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9 | simplrr 526 |
. . . . . . 7
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10 | 0red 7791 |
. . . . . . . 8
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11 | 2, 10 | letri3d 7903 |
. . . . . . 7
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12 | 8, 9, 11 | mpbir2and 929 |
. . . . . 6
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13 | 12 | oveq2d 5798 |
. . . . 5
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14 | 3 | recnd 7818 |
. . . . . 6
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15 | 14 | addid1d 7935 |
. . . . 5
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16 | 13, 7, 15 | 3eqtr3rd 2182 |
. . . 4
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17 | 16, 12 | jca 304 |
. . 3
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18 | 17 | ex 114 |
. 2
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19 | oveq12 5791 |
. . 3
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20 | 00id 7927 |
. . 3
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21 | 19, 20 | eqtrdi 2189 |
. 2
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22 | 18, 21 | impbid1 141 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-pre-ltirr 7756 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-iota 5096 df-fv 5139 df-ov 5785 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 |
This theorem is referenced by: add20i 8278 xnn0xadd0 9680 sumsqeq0 10402 |
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