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Mirrors > Home > MPE Home > Th. List > xnn0lem1lt | Structured version Visualization version GIF version |
Description: Extended nonnegative integer ordering relation. (Contributed by Thierry Arnoux, 30-Jul-2023.) |
Ref | Expression |
---|---|
xnn0lem1lt | ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0lem1lt 11901 | . . 3 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) | |
2 | 1 | adantlr 711 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
3 | nn0re 11759 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ) | |
4 | 3 | rexrd 10542 | . . . . . 6 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℝ*) |
5 | pnfge 12380 | . . . . . 6 ⊢ (𝑀 ∈ ℝ* → 𝑀 ≤ +∞) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ≤ +∞) |
7 | 6 | ad2antrr 722 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → 𝑀 ≤ +∞) |
8 | simpll 763 | . . . . 5 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → 𝑀 ∈ ℕ0) | |
9 | peano2rem 10806 | . . . . 5 ⊢ (𝑀 ∈ ℝ → (𝑀 − 1) ∈ ℝ) | |
10 | ltpnf 12370 | . . . . 5 ⊢ ((𝑀 − 1) ∈ ℝ → (𝑀 − 1) < +∞) | |
11 | 8, 3, 9, 10 | 4syl 19 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → (𝑀 − 1) < +∞) |
12 | 7, 11 | 2thd 266 | . . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → (𝑀 ≤ +∞ ↔ (𝑀 − 1) < +∞)) |
13 | xnn0nnn0pnf 11833 | . . . . 5 ⊢ ((𝑁 ∈ ℕ0* ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) | |
14 | 13 | adantll 710 | . . . 4 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → 𝑁 = +∞) |
15 | 14 | breq2d 4978 | . . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ 𝑀 ≤ +∞)) |
16 | 14 | breq2d 4978 | . . 3 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → ((𝑀 − 1) < 𝑁 ↔ (𝑀 − 1) < +∞)) |
17 | 12, 15, 16 | 3bitr4d 312 | . 2 ⊢ (((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) ∧ ¬ 𝑁 ∈ ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
18 | 2, 17 | pm2.61dan 809 | 1 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0*) → (𝑀 ≤ 𝑁 ↔ (𝑀 − 1) < 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 class class class wbr 4966 (class class class)co 7021 ℝcr 10387 1c1 10389 +∞cpnf 10523 ℝ*cxr 10525 < clt 10526 ≤ cle 10527 − cmin 10722 ℕ0cn0 11750 ℕ0*cxnn0 11820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-n0 11751 df-xnn0 11821 df-z 11835 |
This theorem is referenced by: xnn01gt 30188 drngdimgt0 30625 cusgracyclt3v 32018 |
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