Theorem n0sge0 28489

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 5109	. 2 ⊢ (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s ))
2		breq2 5109	. 2 ⊢ (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚))
3		breq2 5109	. 2 ⊢ (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s )))
4		breq2 5109	. 2 ⊢ (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴))
5		0no 27960	. . 3 ⊢ 0s ∈ No
6		lesid 27889	. . 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 28474	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
10	9	adantr 485	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ∈ No )
11		peano2no 28135	. . . . . 6 ⊢ (𝑚 ∈ No → (𝑚 +s 1s ) ∈ No )
12	9, 11	syl 18	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 1s ) ∈ No )
13	12	adantr 485	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 1s ) ∈ No )
14		simpr 489	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s 𝑚)
15	9	addsridd 28116	. . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) = 𝑚)
16	15	adantr 485	. . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) = 𝑚)
17	5	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s ∈ No )
18		1no 27961	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 1s ∈ No )
20		0lt1s 27963	. . . . . . . . . 10 ⊢ 0s <s 1s
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s <s 1s )
22	17, 19, 21	ltlesd 27895	. . . . . . . 8 ⊢ (⊤ → 0s ≤s 1s )
23	22	mptru 1570	. . . . . . 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 28147	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 1s ↔ (𝑚 +s 0s ) ≤s (𝑚 +s 1s )))
27	23, 26	mpbii 236	. . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
28	27	adantr 485	. . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
29	16, 28	eqbrtrrd 5129	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s ))
30	8, 10, 13, 14, 29	lestrd 27888	. . 3 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s (𝑚 +s 1s ))
31	30	ex 417	. 2 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 𝑚 → 0s ≤s (𝑚 +s 1s )))
32	1, 2, 3, 4, 7, 31	n0sind 28484	1 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563  ⊤wtru 1564   ∈ wcel 2145   class class class wbr 5105  (class class class)co 7400   No csur 27762   <s clts 27763   ≤s cles 27866   0_s c0s 27956   1_s c1s 27957   +_s cadds 28110  ℕ_0scn0s 28463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec2 28100  df-adds 28111  df-n0s 28465

This theorem is referenced by:  nnsgt0  28490  elnns2  28492  nnsge1  28494  n0subs  28514  n0lts1e0  28519  eln0zs  28551  bdaypw2n0bndlem  28614  bdayfinbndlem1  28618  z12bdaylem1  28621