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Mirrors > Home > ILE Home > Th. List > xaddnemnf | Unicode version |
Description: Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddnemnf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnemnf 9594 | . 2 | |
2 | xrnemnf 9594 | . . . 4 | |
3 | rexadd 9665 | . . . . . . 7 | |
4 | readdcl 7770 | . . . . . . 7 | |
5 | 3, 4 | eqeltrd 2217 | . . . . . 6 |
6 | 5 | renemnfd 7841 | . . . . 5 |
7 | oveq2 5790 | . . . . . . 7 | |
8 | rexr 7835 | . . . . . . . 8 | |
9 | renemnf 7838 | . . . . . . . 8 | |
10 | xaddpnf1 9659 | . . . . . . . 8 | |
11 | 8, 9, 10 | syl2anc 409 | . . . . . . 7 |
12 | 7, 11 | sylan9eqr 2195 | . . . . . 6 |
13 | pnfnemnf 7844 | . . . . . . 7 | |
14 | 13 | a1i 9 | . . . . . 6 |
15 | 12, 14 | eqnetrd 2333 | . . . . 5 |
16 | 6, 15 | jaodan 787 | . . . 4 |
17 | 2, 16 | sylan2b 285 | . . 3 |
18 | oveq1 5789 | . . . . 5 | |
19 | xaddpnf2 9660 | . . . . 5 | |
20 | 18, 19 | sylan9eq 2193 | . . . 4 |
21 | 13 | a1i 9 | . . . 4 |
22 | 20, 21 | eqnetrd 2333 | . . 3 |
23 | 17, 22 | jaoian 785 | . 2 |
24 | 1, 23 | sylanb 282 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1332 wcel 1481 wne 2309 (class class class)co 5782 cr 7643 caddc 7647 cpnf 7821 cmnf 7822 cxr 7823 cxad 9587 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-rnegex 7753 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-xadd 9590 |
This theorem is referenced by: xaddass 9682 xlt2add 9693 xadd4d 9698 xleaddadd 9700 |
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