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Mirrors > Home > ILE Home > Th. List > gt0add | Unicode version |
Description: A positive sum must have a positive addend. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by Jim Kingdon, 26-Jan-2020.) |
Ref | Expression |
---|---|
gt0add |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3 948 |
. . 3
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2 | 0red 7586 |
. . . 4
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3 | simp1 946 |
. . . . 5
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4 | simp2 947 |
. . . . 5
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5 | 3, 4 | readdcld 7614 |
. . . 4
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6 | axltwlin 7651 |
. . . 4
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7 | 2, 5, 3, 6 | syl3anc 1181 |
. . 3
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8 | 1, 7 | mpd 13 |
. 2
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9 | 4, 3 | ltaddposd 8103 |
. . 3
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10 | 9 | orbi2d 742 |
. 2
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11 | 8, 10 | mpbird 166 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-i2m1 7547 ax-0id 7550 ax-rnegex 7551 ax-pre-ltwlin 7555 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-xp 4473 df-iota 5014 df-fv 5057 df-ov 5693 df-pnf 7621 df-mnf 7622 df-ltxr 7624 |
This theorem is referenced by: (None) |
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