Theorem n0sge0 28346

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 5104	. 2 ⊢ (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s ))
2		breq2 5104	. 2 ⊢ (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚))
3		breq2 5104	. 2 ⊢ (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s )))
4		breq2 5104	. 2 ⊢ (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴))
5		0no 27817	. . 3 ⊢ 0s ∈ No
6		lesid 27747	. . 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 28331	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
10	9	adantr 480	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ∈ No )
11		peano2no 27992	. . . . . 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 27973	. . . . . 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		1no 27818	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 1s ∈ No )
20		0lt1s 27820	. . . . . . . . . 10 ⊢ 0s <s 1s
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s <s 1s )
22	17, 19, 21	ltlesd 27753	. . . . . . . 8 ⊢ (⊤ → 0s ≤s 1s )
23	22	mptru 1549	. . . . . . 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 28004	. . . . . . 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 5124	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s ))
30	8, 10, 13, 14, 29	lestrd 27746	. . 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 28341	1 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114   class class class wbr 5100  (class class class)co 7368   No csur 27619   <s clts 27620   ≤s cles 27724   0_s c0s 27813   1_s c1s 27814   +_s cadds 27967  ℕ_0scn0s 28320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec2 27957  df-adds 27968  df-n0s 28322

This theorem is referenced by:  nnsgt0  28347  elnns2  28349  nnsge1  28351  n0subs  28371  n0lts1e0  28376  eln0zs  28408  bdaypw2n0bndlem  28471  bdayfinbndlem1  28475  z12bdaylem1  28478