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Mirrors > Home > ILE Home > Th. List > xaddmnf2 | Unicode version |
Description: Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddmnf2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 7965 | . . 3 | |
2 | xaddval 9791 | . . 3 | |
3 | 1, 2 | mpan 422 | . 2 |
4 | mnfnepnf 7964 | . . . . 5 | |
5 | ifnefalse 3536 | . . . . 5 | |
6 | 4, 5 | ax-mp 5 | . . . 4 |
7 | eqid 2170 | . . . . 5 | |
8 | 7 | iftruei 3531 | . . . 4 |
9 | 6, 8 | eqtri 2191 | . . 3 |
10 | ifnefalse 3536 | . . 3 | |
11 | 9, 10 | eqtrid 2215 | . 2 |
12 | 3, 11 | sylan9eq 2223 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 wne 2340 cif 3525 (class class class)co 5851 cc0 7763 caddc 7766 cpnf 7940 cmnf 7941 cxr 7942 cxad 9716 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1re 7857 ax-addrcl 7860 ax-rnegex 7872 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-pnf 7945 df-mnf 7946 df-xr 7947 df-xadd 9719 |
This theorem is referenced by: xaddnepnf 9804 xaddcom 9807 xaddid1 9808 xnegdi 9814 xpncan 9817 xleadd1a 9819 xltadd1 9822 xlt2add 9826 xposdif 9828 xleaddadd 9833 xrmaxadd 11213 |
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