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Mirrors > Home > MPE Home > Th. List > zltlem1 | Structured version Visualization version GIF version |
Description: Integer ordering relation. (Contributed by NM, 13-Nov-2004.) |
Ref | Expression |
---|---|
zltlem1 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2zm 12026 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
2 | zleltp1 12034 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) | |
3 | 1, 2 | sylan2 594 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) |
4 | zcn 11987 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
5 | ax-1cn 10595 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | npcan 10895 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 588 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | adantl 484 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | breq2d 5078 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < ((𝑁 − 1) + 1) ↔ 𝑀 < 𝑁)) |
10 | 3, 9 | bitr2d 282 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 ℂcc 10535 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 − cmin 10870 ℤcz 11982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 |
This theorem is referenced by: nn0ltlem1 12043 nn0lt2 12046 nn0le2is012 12047 nnltlem1 12050 nnm1ge0 12051 zextlt 12057 uzm1 12277 elfzm11 12979 preduz 13030 predfz 13033 elfzo 13041 fzosplitprm1 13148 intfracq 13228 seqf1olem1 13410 seqcoll 13823 isercolllem1 15021 fzm1ndvds 15672 bitscmp 15787 nn0seqcvgd 15914 isprm3 16027 ncoprmlnprm 16068 prmdiveq 16123 4sqlem12 16292 degltlem1 24666 dgreq0 24855 wilthlem1 25645 lgseisenlem2 25952 lgsquadlem1 25956 2lgslem1a1 25965 2sqlem8 26002 crctcshwlkn0lem4 27591 clwlkclwwlklem2a4 27775 clwlkclwwlklem2a 27776 frgrreggt1 28172 bcm1n 30518 smatrcl 31061 ballotlemimin 31763 ballotlemfrcn0 31787 knoppndvlem2 33852 poimirlem2 34909 poimirlem24 34931 fmul01lt1lem2 41886 fourierdlem41 42453 fourierdlem42 42454 fourierdlem50 42461 fourierdlem64 42475 fourierdlem79 42490 etransclem44 42583 etransclem48 42587 pw2m1lepw2m1 44595 fllog2 44648 |
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