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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 12658 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
2 | zleltp1 12666 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) | |
3 | 1, 2 | sylan2 593 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ (𝑁 − 1) ↔ 𝑀 < ((𝑁 − 1) + 1))) |
4 | zcn 12616 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
5 | ax-1cn 11211 | . . . . 5 ⊢ 1 ∈ ℂ | |
6 | npcan 11515 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
7 | 4, 5, 6 | sylancl 586 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑁 − 1) + 1) = 𝑁) |
8 | 7 | adantl 481 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | breq2d 5160 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < ((𝑁 − 1) + 1) ↔ 𝑀 < 𝑁)) |
10 | 3, 9 | bitr2d 280 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 ℤcz 12611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 |
This theorem is referenced by: zltlem1d 12669 nn0ltlem1 12676 nn0lt2 12679 nn0le2is012 12680 nnltlem1 12683 nnm1ge0 12684 zextlt 12690 uzm1 12914 elfzm11 13632 preduz 13687 predfz 13690 elfzo 13698 fzosplitprm1 13813 intfracq 13896 seqf1olem1 14079 seqcoll 14500 isercolllem1 15698 fzm1ndvds 16356 bitscmp 16472 nn0seqcvgd 16604 isprm3 16717 ncoprmlnprm 16762 prmdiveq 16820 4sqlem12 16990 degltlem1 26126 dgreq0 26320 wilthlem1 27126 lgseisenlem2 27435 lgsquadlem1 27439 2lgslem1a1 27448 2sqlem8 27485 crctcshwlkn0lem4 29843 clwlkclwwlklem2a4 30026 clwlkclwwlklem2a 30027 frgrreggt1 30422 bcm1n 32803 ply1degltel 33595 smatrcl 33757 ballotlemimin 34487 ballotlemfrcn0 34511 knoppndvlem2 36496 poimirlem2 37609 poimirlem24 37631 sticksstones10 42137 metakunt7 42193 metakunt21 42207 metakunt22 42208 metakunt24 42210 fmul01lt1lem2 45541 fourierdlem41 46104 fourierdlem42 46105 fourierdlem50 46112 fourierdlem64 46126 fourierdlem79 46141 etransclem44 46234 etransclem48 46238 pw2m1lepw2m1 48366 fllog2 48418 |
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