Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xnn0xadd0 | Unicode version |
Description: The sum of two extended nonnegative integers is iff each of the two extended nonnegative integers is . (Contributed by AV, 14-Dec-2020.) |
Ref | Expression |
---|---|
xnn0xadd0 | NN0* NN0* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxnn0 9170 | . . . 4 NN0* | |
2 | elxnn0 9170 | . . . . . . 7 NN0* | |
3 | nn0re 9114 | . . . . . . . . . . . . 13 | |
4 | nn0re 9114 | . . . . . . . . . . . . 13 | |
5 | rexadd 9779 | . . . . . . . . . . . . 13 | |
6 | 3, 4, 5 | syl2an 287 | . . . . . . . . . . . 12 |
7 | 6 | eqeq1d 2173 | . . . . . . . . . . 11 |
8 | nn0ge0 9130 | . . . . . . . . . . . . 13 | |
9 | 3, 8 | jca 304 | . . . . . . . . . . . 12 |
10 | nn0ge0 9130 | . . . . . . . . . . . . 13 | |
11 | 4, 10 | jca 304 | . . . . . . . . . . . 12 |
12 | add20 8363 | . . . . . . . . . . . 12 | |
13 | 9, 11, 12 | syl2an 287 | . . . . . . . . . . 11 |
14 | 7, 13 | bitrd 187 | . . . . . . . . . 10 |
15 | 14 | biimpd 143 | . . . . . . . . 9 |
16 | 15 | expcom 115 | . . . . . . . 8 |
17 | oveq2 5844 | . . . . . . . . . . . . 13 | |
18 | 17 | eqeq1d 2173 | . . . . . . . . . . . 12 |
19 | 18 | adantr 274 | . . . . . . . . . . 11 |
20 | nn0xnn0 9172 | . . . . . . . . . . . . . 14 NN0* | |
21 | xnn0xrnemnf 9180 | . . . . . . . . . . . . . 14 NN0* | |
22 | xaddpnf1 9773 | . . . . . . . . . . . . . 14 | |
23 | 20, 21, 22 | 3syl 17 | . . . . . . . . . . . . 13 |
24 | 23 | adantl 275 | . . . . . . . . . . . 12 |
25 | 24 | eqeq1d 2173 | . . . . . . . . . . 11 |
26 | 19, 25 | bitrd 187 | . . . . . . . . . 10 |
27 | 0re 7890 | . . . . . . . . . . . . 13 | |
28 | renepnf 7937 | . . . . . . . . . . . . 13 | |
29 | 27, 28 | ax-mp 5 | . . . . . . . . . . . 12 |
30 | 29 | nesymi 2380 | . . . . . . . . . . 11 |
31 | 30 | pm2.21i 636 | . . . . . . . . . 10 |
32 | 26, 31 | syl6bi 162 | . . . . . . . . 9 |
33 | 32 | ex 114 | . . . . . . . 8 |
34 | 16, 33 | jaoi 706 | . . . . . . 7 |
35 | 2, 34 | sylbi 120 | . . . . . 6 NN0* |
36 | 35 | com12 30 | . . . . 5 NN0* |
37 | oveq1 5843 | . . . . . . . . 9 | |
38 | 37 | eqeq1d 2173 | . . . . . . . 8 |
39 | xnn0xrnemnf 9180 | . . . . . . . . . 10 NN0* | |
40 | xaddpnf2 9774 | . . . . . . . . . 10 | |
41 | 39, 40 | syl 14 | . . . . . . . . 9 NN0* |
42 | 41 | eqeq1d 2173 | . . . . . . . 8 NN0* |
43 | 38, 42 | sylan9bb 458 | . . . . . . 7 NN0* |
44 | 43, 31 | syl6bi 162 | . . . . . 6 NN0* |
45 | 44 | ex 114 | . . . . 5 NN0* |
46 | 36, 45 | jaoi 706 | . . . 4 NN0* |
47 | 1, 46 | sylbi 120 | . . 3 NN0* NN0* |
48 | 47 | imp 123 | . 2 NN0* NN0* |
49 | oveq12 5845 | . . 3 | |
50 | 0xr 7936 | . . . 4 | |
51 | xaddid1 9789 | . . . 4 | |
52 | 50, 51 | ax-mp 5 | . . 3 |
53 | 49, 52 | eqtrdi 2213 | . 2 |
54 | 48, 53 | impbid1 141 | 1 NN0* NN0* |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1342 wcel 2135 wne 2334 class class class wbr 3976 (class class class)co 5836 cr 7743 cc0 7744 caddc 7747 cpnf 7921 cmnf 7922 cxr 7923 cle 7925 cn0 9105 NN0*cxnn0 9168 cxad 9697 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-inn 8849 df-n0 9106 df-xnn0 9169 df-xadd 9700 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |