Theorem n0sge0 28408

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 5103	. 2 ⊢ (𝑛 = 0s → ( 0s ≤s 𝑛 ↔ 0s ≤s 0s ))
2		breq2 5103	. 2 ⊢ (𝑛 = 𝑚 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝑚))
3		breq2 5103	. 2 ⊢ (𝑛 = (𝑚 +s 1s ) → ( 0s ≤s 𝑛 ↔ 0s ≤s (𝑚 +s 1s )))
4		breq2 5103	. 2 ⊢ (𝑛 = 𝐴 → ( 0s ≤s 𝑛 ↔ 0s ≤s 𝐴))
5		0no 27879	. . 3 ⊢ 0s ∈ No
6		lesid 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		n0no 28393	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → 𝑚 ∈ No )
10	9	adantr 484	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ∈ No )
11		peano2no 28054	. . . . . 6 ⊢ (𝑚 ∈ No → (𝑚 +s 1s ) ∈ No )
12	9, 11	syl 17	. . . . 5 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 1s ) ∈ No )
13	12	adantr 484	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 1s ) ∈ No )
14		simpr 488	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s 𝑚)
15	9	addsridd 28035	. . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) = 𝑚)
16	15	adantr 484	. . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) = 𝑚)
17	5	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s ∈ No )
18		1no 27880	. . . . . . . . . 10 ⊢ 1s ∈ No
19	18	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 1s ∈ No )
20		0lt1s 27882	. . . . . . . . . 10 ⊢ 0s <s 1s
21	20	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s <s 1s )
22	17, 19, 21	ltlesd 27814	. . . . . . . 8 ⊢ (⊤ → 0s ≤s 1s )
23	22	mptru 1566	. . . . . . 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 28066	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 1s ↔ (𝑚 +s 0s ) ≤s (𝑚 +s 1s )))
27	23, 26	mpbii 235	. . . . . 6 ⊢ (𝑚 ∈ ℕ0s → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
28	27	adantr 484	. . . . 5 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → (𝑚 +s 0s ) ≤s (𝑚 +s 1s ))
29	16, 28	eqbrtrrd 5123	. . . 4 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 𝑚 ≤s (𝑚 +s 1s ))
30	8, 10, 13, 14, 29	lestrd 27807	. . 3 ⊢ ((𝑚 ∈ ℕ0s ∧ 0s ≤s 𝑚) → 0s ≤s (𝑚 +s 1s ))
31	30	ex 416	. 2 ⊢ (𝑚 ∈ ℕ0s → ( 0s ≤s 𝑚 → 0s ≤s (𝑚 +s 1s )))
32	1, 2, 3, 4, 7, 31	n0sind 28403	1 ⊢ (𝐴 ∈ ℕ0s → 0s ≤s 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ⊤wtru 1560   ∈ wcel 2141   class class class wbr 5099  (class class class)co 7392   No csur 27681   <s clts 27682   ≤s cles 27785   0_s c0s 27875   1_s c1s 27876   +_s cadds 28029  ℕ_0scn0s 28382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-1s 27878  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec2 28019  df-adds 28030  df-n0s 28384

This theorem is referenced by:  nnsgt0  28409  elnns2  28411  nnsge1  28413  n0subs  28433  n0lts1e0  28438  eln0zs  28470  bdaypw2n0bndlem  28533  bdayfinbndlem1  28537  z12bdaylem1  28540