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Mirrors > Home > ILE Home > Th. List > leadd1 | Unicode version |
Description: Addition to both sides of 'less than or equal to'. Part of definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 18-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
leadd1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd1 8384 |
. . . 4
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2 | 1 | 3com12 1207 |
. . 3
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3 | 2 | notbid 667 |
. 2
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4 | simp1 997 |
. . 3
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5 | simp2 998 |
. . 3
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6 | 4, 5 | lenltd 8073 |
. 2
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7 | simp3 999 |
. . . 4
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8 | 4, 7 | readdcld 7985 |
. . 3
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9 | 5, 7 | readdcld 7985 |
. . 3
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10 | 8, 9 | lenltd 8073 |
. 2
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11 | 3, 6, 10 | 3bitr4d 220 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-pre-ltadd 7926 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-cnv 4634 df-iota 5178 df-fv 5224 df-ov 5877 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 |
This theorem is referenced by: leadd2 8386 lesubadd 8389 leaddsub 8393 le2add 8399 leadd1i 8458 leadd1d 8494 zleltp1 9306 eluzp1p1 9551 eluzaddi 9552 icoshft 9988 iccshftr 9992 fzen 10040 fzaddel 10056 fznatpl1 10073 fldiv4p1lem1div2 10302 faclbnd6 10719 |
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