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Mirrors > Home > ILE Home > Th. List > xaddmnf1 | Unicode version |
Description: Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddmnf1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 7815 | . . . 4 | |
2 | xaddval 9621 | . . . 4 | |
3 | 1, 2 | mpan2 421 | . . 3 |
4 | 3 | adantr 274 | . 2 |
5 | ifnefalse 3480 | . . 3 | |
6 | mnfnepnf 7814 | . . . . . 6 | |
7 | ifnefalse 3480 | . . . . . 6 | |
8 | 6, 7 | ax-mp 5 | . . . . 5 |
9 | ifnefalse 3480 | . . . . . . 7 | |
10 | 6, 9 | ax-mp 5 | . . . . . 6 |
11 | eqid 2137 | . . . . . . 7 | |
12 | 11 | iftruei 3475 | . . . . . 6 |
13 | 10, 12 | eqtri 2158 | . . . . 5 |
14 | ifeq12 3483 | . . . . 5 | |
15 | 8, 13, 14 | mp2an 422 | . . . 4 |
16 | xrmnfdc 9619 | . . . . 5 DECID | |
17 | ifiddc 3500 | . . . . 5 DECID | |
18 | 16, 17 | syl 14 | . . . 4 |
19 | 15, 18 | syl5eq 2182 | . . 3 |
20 | 5, 19 | sylan9eqr 2192 | . 2 |
21 | 4, 20 | eqtrd 2170 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 819 wceq 1331 wcel 1480 wne 2306 cif 3469 (class class class)co 5767 cc0 7613 caddc 7616 cpnf 7790 cmnf 7791 cxr 7792 cxad 9550 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-xadd 9553 |
This theorem is referenced by: xaddnepnf 9634 xaddcom 9637 xnegdi 9644 xleadd1a 9649 xsubge0 9657 xposdif 9658 xlesubadd 9659 xleaddadd 9663 xblss2ps 12562 xblss2 12563 |
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