Theorem n0sge0 28355

Description: A non-negative integer is greater than or equal to zero. (Contributed by Scott Fenton, 15-Apr-2025.)

Assertion

Ref	Expression
n0sge0	⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)

Proof of Theorem n0sge0

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5083	. 2 ⊢ (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s ))
2		breq2 5083	. 2 ⊢ (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚))
3		breq2 5083	. 2 ⊢ (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s )))
4		breq2 5083	. 2 ⊢ (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴))
5		0no 27826	. . 3 ⊢ 0s ∈ No
6		lesid 27756	. . 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		n0no 28340	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
10	9	adantr 481	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ∈ No )
11		peano2no 28001	. . . . . 6 ⊢ (𝑚 ∈ No → (𝑚 +s 1s ) ∈ No )
12	9, 11	syl 17	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 1s ) ∈ No )
13	12	adantr 481	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 1s ) ∈ No )
14		simpr 485	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s 𝑚)
15	9	addsridd 27982	. . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) = 𝑚)
16	15	adantr 481	. . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) = 𝑚)
17	5	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s ∈ No )
18		1no 27827	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 1s ∈ No )
20		0lt1s 27829	. . . . . . . . . 10 ⊢ 0s <s 1s
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s <s 1s )
22	17, 19, 21	ltlesd 27762	. . . . . . . 8 ⊢ (⊤ → 0s ≤s 1s )
23	22	mptru 1554	. . . . . . 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	leadds2d 28013	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 1s ↔ (𝑚 +s 0s ) ≤s (𝑚 +s 1s )))
27	23, 26	mpbii 234	. . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
28	27	adantr 481	. . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
29	16, 28	eqbrtrrd 5103	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s ))
30	8, 10, 13, 14, 29	lestrd 27755	. . 3 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s (𝑚 +s 1s ))
31	30	ex 413	. 2 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 𝑚 → 0s ≤s (𝑚 +s 1s )))
32	1, 2, 3, 4, 7, 31	n0sind 28350	1 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ⊤wtru 1548   ∈ wcel 2119   class class class wbr 5079  (class class class)co 7363   No csur 27628   <s clts 27629   ≤s cles 27733   0_s c0s 27822   1_s c1s 27823   +_s cadds 27976  ℕ_0scn0s 28329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec2 27966  df-adds 27977  df-n0s 28331

This theorem is referenced by:  nnsgt0  28356  elnns2  28358  nnsge1  28360  n0subs  28380  n0lts1e0  28385  eln0zs  28417  bdaypw2n0bndlem  28480  bdayfinbndlem1  28484  z12bdaylem1  28487