Theorem n0sge0 28359

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 5170	. 2 ⊢ (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s ))
2		breq2 5170	. 2 ⊢ (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚))
3		breq2 5170	. 2 ⊢ (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s )))
4		breq2 5170	. 2 ⊢ (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴))
5		0sno 27889	. . 3 ⊢ 0s ∈ No
6		slerflex 27826	. . 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 28346	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
10	9	adantr 480	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ∈ No )
11		peano2no 28035	. . . . . 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 28016	. . . . . 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 27890	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 1s ∈ No )
20		0slt1s 27892	. . . . . . . . . 10 ⊢ 0s <s 1s
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s <s 1s )
22	17, 19, 21	sltled 27832	. . . . . . . 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 28047	. . . . . . 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 5190	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s ))
30	8, 10, 13, 14, 29	sletrd 27825	. . 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 28355	1 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537  ⊤wtru 1538   ∈ wcel 2108   class class class wbr 5166  (class class class)co 7448   No csur 27702   <s cslt 27703   ≤s csle 27807   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010  ℕ_0scnn0s 28336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011  df-n0s 28338

This theorem is referenced by:  nnsgt0  28360  elnns2  28362  nnsge1  28364  n0subs  28378  eln0zs  28404