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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 9721 | . 2 | |
2 | xrnemnf 9721 | . . . 4 | |
3 | rexadd 9796 | . . . . . . 7 | |
4 | readdcl 7887 | . . . . . . 7 | |
5 | 3, 4 | eqeltrd 2247 | . . . . . 6 |
6 | 5 | renemnfd 7958 | . . . . 5 |
7 | oveq2 5858 | . . . . . . 7 | |
8 | rexr 7952 | . . . . . . . 8 | |
9 | renemnf 7955 | . . . . . . . 8 | |
10 | xaddpnf1 9790 | . . . . . . . 8 | |
11 | 8, 9, 10 | syl2anc 409 | . . . . . . 7 |
12 | 7, 11 | sylan9eqr 2225 | . . . . . 6 |
13 | pnfnemnf 7961 | . . . . . . 7 | |
14 | 13 | a1i 9 | . . . . . 6 |
15 | 12, 14 | eqnetrd 2364 | . . . . 5 |
16 | 6, 15 | jaodan 792 | . . . 4 |
17 | 2, 16 | sylan2b 285 | . . 3 |
18 | oveq1 5857 | . . . . 5 | |
19 | xaddpnf2 9791 | . . . . 5 | |
20 | 18, 19 | sylan9eq 2223 | . . . 4 |
21 | 13 | a1i 9 | . . . 4 |
22 | 20, 21 | eqnetrd 2364 | . . 3 |
23 | 17, 22 | jaoian 790 | . 2 |
24 | 1, 23 | sylanb 282 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 wceq 1348 wcel 2141 wne 2340 (class class class)co 5850 cr 7760 caddc 7764 cpnf 7938 cmnf 7939 cxr 7940 cxad 9714 |
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 7852 ax-resscn 7853 ax-1re 7855 ax-addrcl 7858 ax-rnegex 7870 |
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 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-xadd 9717 |
This theorem is referenced by: xaddass 9813 xlt2add 9824 xadd4d 9829 xleaddadd 9831 |
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