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Mirrors > Home > ILE Home > Th. List > xaddpnf2 | Unicode version |
Description: Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddpnf2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfxr 7987 |
. . 3
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2 | xaddval 9819 |
. . 3
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3 | 1, 2 | mpan 424 |
. 2
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4 | eqid 2177 |
. . . 4
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5 | 4 | iftruei 3540 |
. . 3
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6 | ifnefalse 3545 |
. . 3
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7 | 5, 6 | eqtrid 2222 |
. 2
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8 | 3, 7 | sylan9eq 2230 |
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 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 ax-rnegex 7898 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 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-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-iota 5173 df-fun 5213 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-xadd 9747 |
This theorem is referenced by: xaddnemnf 9831 xaddcom 9835 xaddid1 9836 xnn0xadd0 9841 xnegdi 9842 xaddass 9843 xleadd1a 9847 xltadd1 9850 xposdif 9856 xleaddadd 9861 xrmaxadd 11240 xrbdtri 11255 isxmet2d 13481 |
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