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Mirrors > Home > MPE Home > Th. List > Mathboxes > evenltle | Structured version Visualization version GIF version |
Description: If an even number is greater than another even number, then it is greater than or equal to the other even number plus 2. (Contributed by AV, 25-Dec-2021.) |
Ref | Expression |
---|---|
evenltle | ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ∧ 𝑀 < 𝑁) → (𝑀 + 2) ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evenz 47555 | . . . 4 ⊢ (𝑀 ∈ Even → 𝑀 ∈ ℤ) | |
2 | evenz 47555 | . . . 4 ⊢ (𝑁 ∈ Even → 𝑁 ∈ ℤ) | |
3 | zltp1le 12665 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | |
4 | 1, 2, 3 | syl2anr 597 | . . 3 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
5 | 1 | zred 12720 | . . . . . 6 ⊢ (𝑀 ∈ Even → 𝑀 ∈ ℝ) |
6 | peano2re 11432 | . . . . . 6 ⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝑀 ∈ Even → (𝑀 + 1) ∈ ℝ) |
8 | 2 | zred 12720 | . . . . 5 ⊢ (𝑁 ∈ Even → 𝑁 ∈ ℝ) |
9 | leloe 11345 | . . . . 5 ⊢ (((𝑀 + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 1) ≤ 𝑁 ↔ ((𝑀 + 1) < 𝑁 ∨ (𝑀 + 1) = 𝑁))) | |
10 | 7, 8, 9 | syl2anr 597 | . . . 4 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → ((𝑀 + 1) ≤ 𝑁 ↔ ((𝑀 + 1) < 𝑁 ∨ (𝑀 + 1) = 𝑁))) |
11 | 1 | peano2zd 12723 | . . . . . . 7 ⊢ (𝑀 ∈ Even → (𝑀 + 1) ∈ ℤ) |
12 | zltp1le 12665 | . . . . . . 7 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) < 𝑁 ↔ ((𝑀 + 1) + 1) ≤ 𝑁)) | |
13 | 11, 2, 12 | syl2anr 597 | . . . . . 6 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → ((𝑀 + 1) < 𝑁 ↔ ((𝑀 + 1) + 1) ≤ 𝑁)) |
14 | 1 | zcnd 12721 | . . . . . . . . . 10 ⊢ (𝑀 ∈ Even → 𝑀 ∈ ℂ) |
15 | 14 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → 𝑀 ∈ ℂ) |
16 | add1p1 12515 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℂ → ((𝑀 + 1) + 1) = (𝑀 + 2)) | |
17 | 15, 16 | syl 17 | . . . . . . . 8 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → ((𝑀 + 1) + 1) = (𝑀 + 2)) |
18 | 17 | breq1d 5158 | . . . . . . 7 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → (((𝑀 + 1) + 1) ≤ 𝑁 ↔ (𝑀 + 2) ≤ 𝑁)) |
19 | 18 | biimpd 229 | . . . . . 6 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → (((𝑀 + 1) + 1) ≤ 𝑁 → (𝑀 + 2) ≤ 𝑁)) |
20 | 13, 19 | sylbid 240 | . . . . 5 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → ((𝑀 + 1) < 𝑁 → (𝑀 + 2) ≤ 𝑁)) |
21 | evenp1odd 47565 | . . . . . 6 ⊢ (𝑀 ∈ Even → (𝑀 + 1) ∈ Odd ) | |
22 | zneoALTV 47594 | . . . . . . 7 ⊢ ((𝑁 ∈ Even ∧ (𝑀 + 1) ∈ Odd ) → 𝑁 ≠ (𝑀 + 1)) | |
23 | eqneqall 2949 | . . . . . . . 8 ⊢ (𝑁 = (𝑀 + 1) → (𝑁 ≠ (𝑀 + 1) → (𝑀 + 2) ≤ 𝑁)) | |
24 | 23 | eqcoms 2743 | . . . . . . 7 ⊢ ((𝑀 + 1) = 𝑁 → (𝑁 ≠ (𝑀 + 1) → (𝑀 + 2) ≤ 𝑁)) |
25 | 22, 24 | syl5com 31 | . . . . . 6 ⊢ ((𝑁 ∈ Even ∧ (𝑀 + 1) ∈ Odd ) → ((𝑀 + 1) = 𝑁 → (𝑀 + 2) ≤ 𝑁)) |
26 | 21, 25 | sylan2 593 | . . . . 5 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → ((𝑀 + 1) = 𝑁 → (𝑀 + 2) ≤ 𝑁)) |
27 | 20, 26 | jaod 859 | . . . 4 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → (((𝑀 + 1) < 𝑁 ∨ (𝑀 + 1) = 𝑁) → (𝑀 + 2) ≤ 𝑁)) |
28 | 10, 27 | sylbid 240 | . . 3 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → ((𝑀 + 1) ≤ 𝑁 → (𝑀 + 2) ≤ 𝑁)) |
29 | 4, 28 | sylbid 240 | . 2 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ) → (𝑀 < 𝑁 → (𝑀 + 2) ≤ 𝑁)) |
30 | 29 | 3impia 1116 | 1 ⊢ ((𝑁 ∈ Even ∧ 𝑀 ∈ Even ∧ 𝑀 < 𝑁) → (𝑀 + 2) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 (class class class)co 7431 ℂcc 11151 ℝcr 11152 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 2c2 12319 ℤcz 12611 Even ceven 47549 Odd codd 47550 |
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-rmo 3378 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-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-even 47551 df-odd 47552 |
This theorem is referenced by: mogoldbb 47710 |
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