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Mirrors > Home > ILE Home > Th. List > zaddcllemneg | Unicode version |
Description: Lemma for zaddcl 9094. Special case in which is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zaddcllemneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 982 | . . . . 5 | |
2 | 1 | recnd 7794 | . . . 4 |
3 | 2 | negnegd 8064 | . . 3 |
4 | 3 | oveq2d 5790 | . 2 |
5 | negeq 7955 | . . . . . . . 8 | |
6 | 5 | oveq2d 5790 | . . . . . . 7 |
7 | 6 | eleq1d 2208 | . . . . . 6 |
8 | 7 | imbi2d 229 | . . . . 5 |
9 | negeq 7955 | . . . . . . . 8 | |
10 | 9 | oveq2d 5790 | . . . . . . 7 |
11 | 10 | eleq1d 2208 | . . . . . 6 |
12 | 11 | imbi2d 229 | . . . . 5 |
13 | negeq 7955 | . . . . . . . 8 | |
14 | 13 | oveq2d 5790 | . . . . . . 7 |
15 | 14 | eleq1d 2208 | . . . . . 6 |
16 | 15 | imbi2d 229 | . . . . 5 |
17 | negeq 7955 | . . . . . . . 8 | |
18 | 17 | oveq2d 5790 | . . . . . . 7 |
19 | 18 | eleq1d 2208 | . . . . . 6 |
20 | 19 | imbi2d 229 | . . . . 5 |
21 | zcn 9059 | . . . . . . . 8 | |
22 | 21 | adantr 274 | . . . . . . 7 |
23 | 1cnd 7782 | . . . . . . 7 | |
24 | 22, 23 | negsubd 8079 | . . . . . 6 |
25 | peano2zm 9092 | . . . . . . 7 | |
26 | 25 | adantr 274 | . . . . . 6 |
27 | 24, 26 | eqeltrd 2216 | . . . . 5 |
28 | nncn 8728 | . . . . . . . . . . 11 | |
29 | 28 | ad2antrr 479 | . . . . . . . . . 10 |
30 | 1cnd 7782 | . . . . . . . . . 10 | |
31 | 29, 30 | negdi2d 8087 | . . . . . . . . 9 |
32 | 31 | oveq2d 5790 | . . . . . . . 8 |
33 | 22 | ad2antlr 480 | . . . . . . . . . 10 |
34 | 29 | negcld 8060 | . . . . . . . . . 10 |
35 | 33, 34, 30 | addsubassd 8093 | . . . . . . . . 9 |
36 | peano2zm 9092 | . . . . . . . . . 10 | |
37 | 36 | adantl 275 | . . . . . . . . 9 |
38 | 35, 37 | eqeltrrd 2217 | . . . . . . . 8 |
39 | 32, 38 | eqeltrd 2216 | . . . . . . 7 |
40 | 39 | exp31 361 | . . . . . 6 |
41 | 40 | a2d 26 | . . . . 5 |
42 | 8, 12, 16, 20, 27, 41 | nnind 8736 | . . . 4 |
43 | 42 | impcom 124 | . . 3 |
44 | 43 | 3impa 1176 | . 2 |
45 | 4, 44 | eqeltrrd 2217 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 (class class class)co 5774 cc 7618 cr 7619 c1 7621 caddc 7623 cmin 7933 cneg 7934 cn 8720 cz 9054 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 |
This theorem is referenced by: zaddcl 9094 |
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