Theorem eucliddivs 28435

Description: Euclid's division lemma for surreal numbers. (Contributed by Scott Fenton, 8-Nov-2025.)

Assertion

Ref	Expression
eucliddivs	⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))

Distinct variable groups:   𝐴,𝑝,𝑞   𝐵,𝑝,𝑞

Proof of Theorem eucliddivs

Dummy variables 𝑎 𝑚 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2756	. . . . . 6 ⊢ (𝑚 = 0s → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 0s = ((𝐵 ·s 𝑝) +s 𝑞)))
2	1	anbi1d 639	. . . . 5 ⊢ (𝑚 = 0s → ((𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
3	2	2rexbidv 3217	. . . 4 ⊢ (𝑚 = 0s → (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
4	3	imbi2d 342	. . 3 ⊢ (𝑚 = 0s → ((𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))))
5		eqeq1 2756	. . . . . 6 ⊢ (𝑚 = 𝑎 → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 𝑎 = ((𝐵 ·s 𝑝) +s 𝑞)))
6	5	anbi1d 639	. . . . 5 ⊢ (𝑚 = 𝑎 → ((𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
7	6	2rexbidv 3217	. . . 4 ⊢ (𝑚 = 𝑎 → (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
8	7	imbi2d 342	. . 3 ⊢ (𝑚 = 𝑎 → ((𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))))
9		eqeq1 2756	. . . . . . 7 ⊢ (𝑚 = (𝑎 +s 1s ) → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ (𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞)))
10	9	anbi1d 639	. . . . . 6 ⊢ (𝑚 = (𝑎 +s 1s ) → ((𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ((𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
11	10	2rexbidv 3217	. . . . 5 ⊢ (𝑚 = (𝑎 +s 1s ) → (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
12		oveq2 7389	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → (𝐵 ·s 𝑝) = (𝐵 ·s 𝑟))
13	12	oveq1d 7396	. . . . . . . 8 ⊢ (𝑝 = 𝑟 → ((𝐵 ·s 𝑝) +s 𝑞) = ((𝐵 ·s 𝑟) +s 𝑞))
14	13	eqeq2d 2763	. . . . . . 7 ⊢ (𝑝 = 𝑟 → ((𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞) ↔ (𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑞)))
15	14	anbi1d 639	. . . . . 6 ⊢ (𝑝 = 𝑟 → (((𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑞) ∧ 𝑞 <s 𝐵)))
16		oveq2 7389	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝐵 ·s 𝑟) +s 𝑞) = ((𝐵 ·s 𝑟) +s 𝑠))
17	16	eqeq2d 2763	. . . . . . 7 ⊢ (𝑞 = 𝑠 → ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑞) ↔ (𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠)))
18		breq1 5093	. . . . . . 7 ⊢ (𝑞 = 𝑠 → (𝑞 <s 𝐵 ↔ 𝑠 <s 𝐵))
19	17, 18	anbi12d 640	. . . . . 6 ⊢ (𝑞 = 𝑠 → (((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
20	15, 19	cbvrex2vw 3235	. . . . 5 ⊢ (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))
21	11, 20	bitrdi 289	. . . 4 ⊢ (𝑚 = (𝑎 +s 1s ) → (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
22	21	imbi2d 342	. . 3 ⊢ (𝑚 = (𝑎 +s 1s ) → ((𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕs → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))))
23		eqeq1 2756	. . . . . 6 ⊢ (𝑚 = 𝐴 → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 𝐴 = ((𝐵 ·s 𝑝) +s 𝑞)))
24	23	anbi1d 639	. . . . 5 ⊢ (𝑚 = 𝐴 → ((𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
25	24	2rexbidv 3217	. . . 4 ⊢ (𝑚 = 𝐴 → (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
26	25	imbi2d 342	. . 3 ⊢ (𝑚 = 𝐴 → ((𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))))
27		nnno 28383	. . . . . . 7 ⊢ (𝐵 ∈ ℕs → 𝐵 ∈ No )
28		muls01 28171	. . . . . . 7 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s )
29	27, 28	syl 17	. . . . . 6 ⊢ (𝐵 ∈ ℕs → (𝐵 ·s 0s ) = 0s )
30	29	oveq1d 7396	. . . . 5 ⊢ (𝐵 ∈ ℕs → ((𝐵 ·s 0s ) +s 0s ) = ( 0s +s 0s ))
31		0no 27868	. . . . . 6 ⊢ 0s ∈ No
32		addslid 28027	. . . . . 6 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
33	31, 32	ax-mp 5	. . . . 5 ⊢ ( 0s +s 0s ) = 0s
34	30, 33	eqtr2di 2804	. . . 4 ⊢ (𝐵 ∈ ℕs → 0s = ((𝐵 ·s 0s ) +s 0s ))
35		nnsgt0 28398	. . . 4 ⊢ (𝐵 ∈ ℕs → 0s <s 𝐵)
36		0n0s 28388	. . . . 5 ⊢ 0s ∈ ℕ0s
37		oveq2 7389	. . . . . . . . 9 ⊢ (𝑝 = 0s → (𝐵 ·s 𝑝) = (𝐵 ·s 0s ))
38	37	oveq1d 7396	. . . . . . . 8 ⊢ (𝑝 = 0s → ((𝐵 ·s 𝑝) +s 𝑞) = ((𝐵 ·s 0s ) +s 𝑞))
39	38	eqeq2d 2763	. . . . . . 7 ⊢ (𝑝 = 0s → ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 0s = ((𝐵 ·s 0s ) +s 𝑞)))
40	39	anbi1d 639	. . . . . 6 ⊢ (𝑝 = 0s → (( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0s = ((𝐵 ·s 0s ) +s 𝑞) ∧ 𝑞 <s 𝐵)))
41		oveq2 7389	. . . . . . . 8 ⊢ (𝑞 = 0s → ((𝐵 ·s 0s ) +s 𝑞) = ((𝐵 ·s 0s ) +s 0s ))
42	41	eqeq2d 2763	. . . . . . 7 ⊢ (𝑞 = 0s → ( 0s = ((𝐵 ·s 0s ) +s 𝑞) ↔ 0s = ((𝐵 ·s 0s ) +s 0s )))
43		breq1 5093	. . . . . . 7 ⊢ (𝑞 = 0s → (𝑞 <s 𝐵 ↔ 0s <s 𝐵))
44	42, 43	anbi12d 640	. . . . . 6 ⊢ (𝑞 = 0s → (( 0s = ((𝐵 ·s 0s ) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0s = ((𝐵 ·s 0s ) +s 0s ) ∧ 0s <s 𝐵)))
45	40, 44	rspc2ev 3585	. . . . 5 ⊢ (( 0s ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ ( 0s = ((𝐵 ·s 0s ) +s 0s ) ∧ 0s <s 𝐵)) → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))
46	36, 36, 45	mp3an12 1462	. . . 4 ⊢ (( 0s = ((𝐵 ·s 0s ) +s 0s ) ∧ 0s <s 𝐵) → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))
47	34, 35, 46	syl2anc 592	. . 3 ⊢ (𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))
48		simprr 780	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑞 ∈ ℕ0s)
49		simplr 776	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝐵 ∈ ℕs)
50		nnm1n0s 28434	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕs → (𝐵 -s 1s ) ∈ ℕ0s)
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 -s 1s ) ∈ ℕ0s)
52		n0lesltp1 28425	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℕ0s ∧ (𝐵 -s 1s ) ∈ ℕ0s) → (𝑞 ≤s (𝐵 -s 1s ) ↔ 𝑞 <s ((𝐵 -s 1s ) +s 1s )))
53	48, 51, 52	syl2anc 592	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 ≤s (𝐵 -s 1s ) ↔ 𝑞 <s ((𝐵 -s 1s ) +s 1s )))
54	48	n0nod 28384	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑞 ∈ No )
55	51	n0nod 28384	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 -s 1s ) ∈ No )
56		lesloe 27784	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ No ∧ (𝐵 -s 1s ) ∈ No ) → (𝑞 ≤s (𝐵 -s 1s ) ↔ (𝑞 <s (𝐵 -s 1s ) ∨ 𝑞 = (𝐵 -s 1s ))))
57	54, 55, 56	syl2anc 592	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 ≤s (𝐵 -s 1s ) ↔ (𝑞 <s (𝐵 -s 1s ) ∨ 𝑞 = (𝐵 -s 1s ))))
58	49	nnnod 28385	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝐵 ∈ No )
59		1no 27869	. . . . . . . . . . . 12 ⊢ 1s ∈ No
60		npcans 28134	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 1s ∈ No ) → ((𝐵 -s 1s ) +s 1s ) = 𝐵)
61	58, 59, 60	sylancl 594	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝐵 -s 1s ) +s 1s ) = 𝐵)
62	61	breq2d 5102	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 <s ((𝐵 -s 1s ) +s 1s ) ↔ 𝑞 <s 𝐵))
63	53, 57, 62	3bitr3rd 312	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 <s 𝐵 ↔ (𝑞 <s (𝐵 -s 1s ) ∨ 𝑞 = (𝐵 -s 1s ))))
64		simplrl 784	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 -s 1s )) → 𝑝 ∈ ℕ0s)
65		simplrr 785	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 -s 1s )) → 𝑞 ∈ ℕ0s)
66		peano2n0s 28389	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ0s → (𝑞 +s 1s ) ∈ ℕ0s)
67	65, 66	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 -s 1s )) → (𝑞 +s 1s ) ∈ ℕ0s)
68	49	nnn0sd 28387	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝐵 ∈ ℕ0s)
69		simprl 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
70		n0mulscl 28404	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝐵 ·s 𝑝) ∈ ℕ0s)
71	68, 69, 70	syl2anc 592	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s 𝑝) ∈ ℕ0s)
72	71	n0nod 28384	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s 𝑝) ∈ No )
73	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 1s ∈ No )
74	72, 54, 73	addsassd 28065	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s (𝑞 +s 1s )))
75	74	adantr 483	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 -s 1s )) → (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s (𝑞 +s 1s )))
76	54, 73, 58	ltaddsubsd 28150	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝑞 +s 1s ) <s 𝐵 ↔ 𝑞 <s (𝐵 -s 1s )))
77	76	biimpar 480	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 -s 1s )) → (𝑞 +s 1s ) <s 𝐵)
78		oveq2 7389	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑝 → (𝐵 ·s 𝑟) = (𝐵 ·s 𝑝))
79	78	oveq1d 7396	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑝 → ((𝐵 ·s 𝑟) +s 𝑠) = ((𝐵 ·s 𝑝) +s 𝑠))
80	79	eqeq2d 2763	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑝 → ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ↔ (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑠)))
81	80	anbi1d 639	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑝 → (((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑠) ∧ 𝑠 <s 𝐵)))
82		oveq2 7389	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑞 +s 1s ) → ((𝐵 ·s 𝑝) +s 𝑠) = ((𝐵 ·s 𝑝) +s (𝑞 +s 1s )))
83	82	eqeq2d 2763	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑞 +s 1s ) → ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑠) ↔ (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s (𝑞 +s 1s ))))
84		breq1 5093	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑞 +s 1s ) → (𝑠 <s 𝐵 ↔ (𝑞 +s 1s ) <s 𝐵))
85	83, 84	anbi12d 640	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑞 +s 1s ) → (((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s (𝑞 +s 1s )) ∧ (𝑞 +s 1s ) <s 𝐵)))
86	81, 85	rspc2ev 3585	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ0s ∧ (𝑞 +s 1s ) ∈ ℕ0s ∧ ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑝) +s (𝑞 +s 1s )) ∧ (𝑞 +s 1s ) <s 𝐵)) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))
87	64, 67, 75, 77, 86	syl112anc 1385	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 -s 1s )) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))
88	87	ex 415	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 <s (𝐵 -s 1s ) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
89		peano2n0s 28389	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ0s → (𝑝 +s 1s ) ∈ ℕ0s)
90	69, 89	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑝 +s 1s ) ∈ ℕ0s)
91	58	mulsridd 28173	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s 1s ) = 𝐵)
92	91	oveq2d 7397	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝐵 ·s 𝑝) +s (𝐵 ·s 1s )) = ((𝐵 ·s 𝑝) +s 𝐵))
93	69	n0nod 28384	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑝 ∈ No )
94	58, 93, 73	addsdid 28215	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s (𝑝 +s 1s )) = ((𝐵 ·s 𝑝) +s (𝐵 ·s 1s )))
95	61	oveq2d 7397	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝐵 ·s 𝑝) +s ((𝐵 -s 1s ) +s 1s )) = ((𝐵 ·s 𝑝) +s 𝐵))
96	92, 94, 95	3eqtr4rd 2798	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝐵 ·s 𝑝) +s ((𝐵 -s 1s ) +s 1s )) = (𝐵 ·s (𝑝 +s 1s )))
97	72, 55, 73	addsassd 28065	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑝) +s ((𝐵 -s 1s ) +s 1s )))
98		peano2no 28043	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ No → (𝑝 +s 1s ) ∈ No )
99	93, 98	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑝 +s 1s ) ∈ No )
100	58, 99	mulscld 28194	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s (𝑝 +s 1s )) ∈ No )
101	100	addsridd 28024	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝐵 ·s (𝑝 +s 1s )) +s 0s ) = (𝐵 ·s (𝑝 +s 1s )))
102	96, 97, 101	3eqtr4d 2797	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 0s ))
103	49, 35	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 0s <s 𝐵)
104		oveq2 7389	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (𝑝 +s 1s ) → (𝐵 ·s 𝑟) = (𝐵 ·s (𝑝 +s 1s )))
105	104	oveq1d 7396	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝑝 +s 1s ) → ((𝐵 ·s 𝑟) +s 𝑠) = ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠))
106	105	eqeq2d 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑝 +s 1s ) → ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ↔ (((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠)))
107	106	anbi1d 639	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑝 +s 1s ) → (((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠) ∧ 𝑠 <s 𝐵)))
108		oveq2 7389	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 0s → ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠) = ((𝐵 ·s (𝑝 +s 1s )) +s 0s ))
109	108	eqeq2d 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0s → ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠) ↔ (((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 0s )))
110		breq1 5093	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0s → (𝑠 <s 𝐵 ↔ 0s <s 𝐵))
111	109, 110	anbi12d 640	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0s → (((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 0s ) ∧ 0s <s 𝐵)))
112	107, 111	rspc2ev 3585	. . . . . . . . . . . . 13 ⊢ (((𝑝 +s 1s ) ∈ ℕ0s ∧ 0s ∈ ℕ0s ∧ ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 0s ) ∧ 0s <s 𝐵)) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))
113	36, 112	mp3an2 1460	. . . . . . . . . . . 12 ⊢ (((𝑝 +s 1s ) ∈ ℕ0s ∧ ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s (𝑝 +s 1s )) +s 0s ) ∧ 0s <s 𝐵)) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))
114	90, 102, 103, 113	syl12anc 845	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))
115		oveq2 7389	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐵 -s 1s ) → ((𝐵 ·s 𝑝) +s 𝑞) = ((𝐵 ·s 𝑝) +s (𝐵 -s 1s )))
116	115	oveq1d 7396	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝐵 -s 1s ) → (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = (((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ))
117	116	eqeq1d 2754	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝐵 -s 1s ) → ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ↔ (((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠)))
118	117	anbi1d 639	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝐵 -s 1s ) → (((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
119	118	2rexbidv 3217	. . . . . . . . . . 11 ⊢ (𝑞 = (𝐵 -s 1s ) → (∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s (𝐵 -s 1s )) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
120	114, 119	syl5ibrcom 249	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 = (𝐵 -s 1s ) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
121	88, 120	jaod 868	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝑞 <s (𝐵 -s 1s ) ∨ 𝑞 = (𝐵 -s 1s )) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
122	63, 121	sylbid 242	. . . . . . . 8 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
123		oveq1 7388	. . . . . . . . . . . 12 ⊢ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) → (𝑎 +s 1s ) = (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ))
124	123	eqeq1d 2754	. . . . . . . . . . 11 ⊢ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) → ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ↔ (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠)))
125	124	anbi1d 639	. . . . . . . . . 10 ⊢ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) → (((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
126	125	2rexbidv 3217	. . . . . . . . 9 ⊢ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) → (∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
127	126	imbi2d 342	. . . . . . . 8 ⊢ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) → ((𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)) ↔ (𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))))
128	122, 127	syl5ibrcom 249	. . . . . . 7 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) → (𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))))
129	128	impd 413	. . . . . 6 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
130	129	rexlimdvva 3209	. . . . 5 ⊢ ((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) → (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵)))
131	130	ex 415	. . . 4 ⊢ (𝑎 ∈ ℕ0s → (𝐵 ∈ ℕs → (∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))))
132	131	a2d 29	. . 3 ⊢ (𝑎 ∈ ℕ0s → ((𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)) → (𝐵 ∈ ℕs → ∃𝑟 ∈ ℕ0s ∃𝑠 ∈ ℕ0s ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠) ∧ 𝑠 <s 𝐵))))
133	4, 8, 22, 26, 47, 132	n0sind 28392	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
134	133	imp 409	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 856   = wceq 1550   ∈ wcel 2132  ∃wrex 3076   class class class wbr 5090  (class class class)co 7381   No csur 27670   <s clts 27671   ≤s cles 27774   0_s c0s 27864   1_s c1s 27865   +_s cadds 28018   -_s csubs 28079   ·_s cmuls 28165  ℕ_0scn0s 28371  ℕ_scnns 28372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-ot 4581  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-nadd 8620  df-no 27673  df-lts 27674  df-bday 27675  df-les 27775  df-slts 27817  df-cuts 27819  df-0s 27866  df-1s 27867  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27997  df-norec2 28008  df-adds 28019  df-negs 28080  df-subs 28081  df-muls 28166  df-n0s 28373  df-nns 28374

This theorem is referenced by:  z12sge0  28542