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Mirrors > Home > MPE Home > Th. List > n0sge0 | Structured version Visualization version GIF version |
Description: A non-negative integer is greater than or equal to zero. (Contributed by Scott Fenton, 15-Apr-2025.) |
Ref | Expression |
---|---|
n0sge0 | ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5152 | . 2 ⊢ (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s )) | |
2 | breq2 5152 | . 2 ⊢ (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚)) | |
3 | breq2 5152 | . 2 ⊢ (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s ))) | |
4 | breq2 5152 | . 2 ⊢ (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴)) | |
5 | 0sno 27886 | . . 3 ⊢ 0s ∈ No | |
6 | slerflex 27823 | . . 3 ⊢ ( 0s ∈ No → 0s ≤s 0s ) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ 0s ≤s 0s |
8 | 5 | a1i 11 | . . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ∈ No ) |
9 | n0sno 28343 | . . . . 5 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No ) | |
10 | 9 | adantr 480 | . . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ∈ No ) |
11 | peano2no 28032 | . . . . . 6 ⊢ (𝑚 ∈ No → (𝑚 +s 1s ) ∈ No ) | |
12 | 9, 11 | syl 17 | . . . . 5 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 1s ) ∈ No ) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 1s ) ∈ No ) |
14 | simpr 484 | . . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s 𝑚) | |
15 | 9 | addsridd 28013 | . . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) = 𝑚) |
16 | 15 | adantr 480 | . . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) = 𝑚) |
17 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 0s ∈ No ) |
18 | 1sno 27887 | . . . . . . . . . 10 ⊢ 1s ∈ No | |
19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 1s ∈ No ) |
20 | 0slt1s 27889 | . . . . . . . . . 10 ⊢ 0s <s 1s | |
21 | 20 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 0s <s 1s ) |
22 | 17, 19, 21 | sltled 27829 | . . . . . . . 8 ⊢ (⊤ → 0s ≤s 1s ) |
23 | 22 | mptru 1544 | . . . . . . 7 ⊢ 0s ≤s 1s |
24 | 5 | a1i 11 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ0s → 0s ∈ No ) |
25 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑚 ∈ ℕ0s → 1s ∈ No ) |
26 | 24, 25, 9 | sleadd2d 28044 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 1s ↔ (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))) |
27 | 23, 26 | mpbii 233 | . . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) ≤s (𝑚 +s 1s )) |
28 | 27 | adantr 480 | . . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) ≤s (𝑚 +s 1s )) |
29 | 16, 28 | eqbrtrrd 5172 | . . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s )) |
30 | 8, 10, 13, 14, 29 | sletrd 27822 | . . 3 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s (𝑚 +s 1s )) |
31 | 30 | ex 412 | . 2 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 𝑚 → 0s ≤s (𝑚 +s 1s ))) |
32 | 1, 2, 3, 4, 7, 31 | n0sind 28352 | 1 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 No csur 27699 <s cslt 27700 ≤s csle 27804 0s c0s 27882 1s c1s 27883 +s cadds 28007 ℕ0scnn0s 28333 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
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-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-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 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-se 5642 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-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-1s 27885 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec2 27997 df-adds 28008 df-n0s 28335 |
This theorem is referenced by: nnsgt0 28357 elnns2 28359 nnsge1 28361 n0subs 28375 eln0zs 28401 |
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