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Mirrors > Home > ILE Home > Th. List > xaddnepnf | Unicode version |
Description: Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddnepnf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnepnf 9735 | . 2 | |
2 | xrnepnf 9735 | . . . 4 | |
3 | rexadd 9809 | . . . . . . 7 | |
4 | readdcl 7900 | . . . . . . 7 | |
5 | 3, 4 | eqeltrd 2247 | . . . . . 6 |
6 | 5 | renepnfd 7970 | . . . . 5 |
7 | oveq2 5861 | . . . . . . 7 | |
8 | rexr 7965 | . . . . . . . 8 | |
9 | renepnf 7967 | . . . . . . . 8 | |
10 | xaddmnf1 9805 | . . . . . . . 8 | |
11 | 8, 9, 10 | syl2anc 409 | . . . . . . 7 |
12 | 7, 11 | sylan9eqr 2225 | . . . . . 6 |
13 | mnfnepnf 7975 | . . . . . . 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 5860 | . . . . 5 | |
19 | xaddmnf2 9806 | . . . . 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 5853 cr 7773 caddc 7777 cpnf 7951 cmnf 7952 cxr 7953 cxad 9727 |
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 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 ax-rnegex 7883 |
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 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-xadd 9730 |
This theorem is referenced by: xlt2add 9837 |
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