| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5147 | . 2 ⊢ (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s )) | |
| 2 | breq2 5147 | . 2 ⊢ (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚)) | |
| 3 | breq2 5147 | . 2 ⊢ (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s ))) | |
| 4 | breq2 5147 | . 2 ⊢ (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴)) | |
| 5 | 0sno 27871 | . . 3 ⊢ 0s ∈ No | |
| 6 | slerflex 27808 | . . 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 28328 | . . . . 5 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No ) | |
| 10 | 9 | adantr 480 | . . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ∈ No ) |
| 11 | peano2no 28017 | . . . . . 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 27998 | . . . . . 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 27872 | . . . . . . . . . 10 ⊢ 1s ∈ No | |
| 19 | 18 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 1s ∈ No ) |
| 20 | 0slt1s 27874 | . . . . . . . . . 10 ⊢ 0s <s 1s | |
| 21 | 20 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 0s <s 1s ) |
| 22 | 17, 19, 21 | sltled 27814 | . . . . . . . 8 ⊢ (⊤ → 0s ≤s 1s ) |
| 23 | 22 | mptru 1547 | . . . . . . 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 28029 | . . . . . . 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 5167 | . . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s )) |
| 30 | 8, 10, 13, 14, 29 | sletrd 27807 | . . 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 28337 | 1 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 No csur 27684 <s cslt 27685 ≤s csle 27789 0s c0s 27867 1s c1s 27868 +s cadds 27992 ℕ0scnn0s 28318 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-1s 27870 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec2 27982 df-adds 27993 df-n0s 28320 |
| This theorem is referenced by: nnsgt0 28342 elnns2 28344 nnsge1 28346 n0subs 28360 eln0zs 28386 |
| Copyright terms: Public domain | W3C validator |