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Theorem eucliddivs 28376

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

**Assertion**
Ref	Expression
eucliddivs	⊢ ((𝐴 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (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 2741	. . . . . 6 ⊢ (𝑚 = 0_s → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞)))
2	1	anbi1d 632	. . . . 5 ⊢ (𝑚 = 0_s → ((𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
3	2	2rexbidv 3202	. . . 4 ⊢ (𝑚 = 0_s → (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
4	3	imbi2d 340	. . 3 ⊢ (𝑚 = 0_s → ((𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))))
5		eqeq1 2741	. . . . . 6 ⊢ (𝑚 = 𝑎 → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞)))
6	5	anbi1d 632	. . . . 5 ⊢ (𝑚 = 𝑎 → ((𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
7	6	2rexbidv 3202	. . . 4 ⊢ (𝑚 = 𝑎 → (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
8	7	imbi2d 340	. . 3 ⊢ (𝑚 = 𝑎 → ((𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))))
9		eqeq1 2741	. . . . . . 7 ⊢ (𝑚 = (𝑎 +_s 1_s ) → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ (𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞)))
10	9	anbi1d 632	. . . . . 6 ⊢ (𝑚 = (𝑎 +_s 1_s ) → ((𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
11	10	2rexbidv 3202	. . . . 5 ⊢ (𝑚 = (𝑎 +_s 1_s ) → (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
12		oveq2 7368	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → (𝐵 ·_s 𝑝) = (𝐵 ·_s 𝑟))
13	12	oveq1d 7375	. . . . . . . 8 ⊢ (𝑝 = 𝑟 → ((𝐵 ·_s 𝑝) +_s 𝑞) = ((𝐵 ·_s 𝑟) +_s 𝑞))
14	13	eqeq2d 2748	. . . . . . 7 ⊢ (𝑝 = 𝑟 → ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ (𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑞)))
15	14	anbi1d 632	. . . . . 6 ⊢ (𝑝 = 𝑟 → (((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
16		oveq2 7368	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝐵 ·_s 𝑟) +_s 𝑞) = ((𝐵 ·_s 𝑟) +_s 𝑠))
17	16	eqeq2d 2748	. . . . . . 7 ⊢ (𝑞 = 𝑠 → ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑞) ↔ (𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠)))
18		breq1 5102	. . . . . . 7 ⊢ (𝑞 = 𝑠 → (𝑞 <s 𝐵 ↔ 𝑠 <s 𝐵))
19	17, 18	anbi12d 633	. . . . . 6 ⊢ (𝑞 = 𝑠 → (((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
20	15, 19	cbvrex2vw 3220	. . . . 5 ⊢ (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))
21	11, 20	bitrdi 287	. . . 4 ⊢ (𝑚 = (𝑎 +_s 1_s ) → (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
22	21	imbi2d 340	. . 3 ⊢ (𝑚 = (𝑎 +_s 1_s ) → ((𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕ_s → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))))
23		eqeq1 2741	. . . . . 6 ⊢ (𝑚 = 𝐴 → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞)))
24	23	anbi1d 632	. . . . 5 ⊢ (𝑚 = 𝐴 → ((𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
25	24	2rexbidv 3202	. . . 4 ⊢ (𝑚 = 𝐴 → (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
26	25	imbi2d 340	. . 3 ⊢ (𝑚 = 𝐴 → ((𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)) ↔ (𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))))
27		nnno 28324	. . . . . . 7 ⊢ (𝐵 ∈ ℕ_s → 𝐵 ∈ No )
28		muls01 28112	. . . . . . 7 ⊢ (𝐵 ∈ No → (𝐵 ·_s 0_s ) = 0_s )
29	27, 28	syl 17	. . . . . 6 ⊢ (𝐵 ∈ ℕ_s → (𝐵 ·_s 0_s ) = 0_s )
30	29	oveq1d 7375	. . . . 5 ⊢ (𝐵 ∈ ℕ_s → ((𝐵 ·_s 0_s ) +_s 0_s ) = ( 0_s +_s 0_s ))
31		0no 27809	. . . . . 6 ⊢ 0_s ∈ No
32		addslid 27968	. . . . . 6 ⊢ ( 0_s ∈ No → ( 0_s +_s 0_s ) = 0_s )
33	31, 32	ax-mp 5	. . . . 5 ⊢ ( 0_s +_s 0_s ) = 0_s
34	30, 33	eqtr2di 2789	. . . 4 ⊢ (𝐵 ∈ ℕ_s → 0_s = ((𝐵 ·_s 0_s ) +_s 0_s ))
35		nnsgt0 28339	. . . 4 ⊢ (𝐵 ∈ ℕ_s → 0_s <s 𝐵)
36		0n0s 28329	. . . . 5 ⊢ 0_s ∈ ℕ_0s
37		oveq2 7368	. . . . . . . . 9 ⊢ (𝑝 = 0_s → (𝐵 ·_s 𝑝) = (𝐵 ·_s 0_s ))
38	37	oveq1d 7375	. . . . . . . 8 ⊢ (𝑝 = 0_s → ((𝐵 ·_s 𝑝) +_s 𝑞) = ((𝐵 ·_s 0_s ) +_s 𝑞))
39	38	eqeq2d 2748	. . . . . . 7 ⊢ (𝑝 = 0_s → ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 0_s = ((𝐵 ·_s 0_s ) +_s 𝑞)))
40	39	anbi1d 632	. . . . . 6 ⊢ (𝑝 = 0_s → (( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0_s = ((𝐵 ·_s 0_s ) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
41		oveq2 7368	. . . . . . . 8 ⊢ (𝑞 = 0_s → ((𝐵 ·_s 0_s ) +_s 𝑞) = ((𝐵 ·_s 0_s ) +_s 0_s ))
42	41	eqeq2d 2748	. . . . . . 7 ⊢ (𝑞 = 0_s → ( 0_s = ((𝐵 ·_s 0_s ) +_s 𝑞) ↔ 0_s = ((𝐵 ·_s 0_s ) +_s 0_s )))
43		breq1 5102	. . . . . . 7 ⊢ (𝑞 = 0_s → (𝑞 <s 𝐵 ↔ 0_s <s 𝐵))
44	42, 43	anbi12d 633	. . . . . 6 ⊢ (𝑞 = 0_s → (( 0_s = ((𝐵 ·_s 0_s ) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0_s = ((𝐵 ·_s 0_s ) +_s 0_s ) ∧ 0_s <s 𝐵)))
45	40, 44	rspc2ev 3590	. . . . 5 ⊢ (( 0_s ∈ ℕ_0s ∧ 0_s ∈ ℕ_0s ∧ ( 0_s = ((𝐵 ·_s 0_s ) +_s 0_s ) ∧ 0_s <s 𝐵)) → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))
46	36, 36, 45	mp3an12 1454	. . . 4 ⊢ (( 0_s = ((𝐵 ·_s 0_s ) +_s 0_s ) ∧ 0_s <s 𝐵) → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))
47	34, 35, 46	syl2anc 585	. . 3 ⊢ (𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))
48		simprr 773	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑞 ∈ ℕ_0s)
49		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝐵 ∈ ℕ_s)
50		nnm1n0s 28375	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ_s → (𝐵 -_s 1_s ) ∈ ℕ_0s)
51	49, 50	syl 17	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 -_s 1_s ) ∈ ℕ_0s)
52		n0lesltp1 28366	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℕ_0s ∧ (𝐵 -_s 1_s ) ∈ ℕ_0s) → (𝑞 ≤s (𝐵 -_s 1_s ) ↔ 𝑞 <s ((𝐵 -_s 1_s ) +_s 1_s )))
53	48, 51, 52	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑞 ≤s (𝐵 -_s 1_s ) ↔ 𝑞 <s ((𝐵 -_s 1_s ) +_s 1_s )))
54	48	n0nod 28325	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑞 ∈ No )
55	51	n0nod 28325	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 -_s 1_s ) ∈ No )
56		lesloe 27726	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ No ∧ (𝐵 -_s 1_s ) ∈ No ) → (𝑞 ≤s (𝐵 -_s 1_s ) ↔ (𝑞 <s (𝐵 -_s 1_s ) ∨ 𝑞 = (𝐵 -_s 1_s ))))
57	54, 55, 56	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑞 ≤s (𝐵 -_s 1_s ) ↔ (𝑞 <s (𝐵 -_s 1_s ) ∨ 𝑞 = (𝐵 -_s 1_s ))))
58	49	nnnod 28326	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝐵 ∈ No )
59		1no 27810	. . . . . . . . . . . 12 ⊢ 1_s ∈ No
60		npcans 28075	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 1_s ∈ No ) → ((𝐵 -_s 1_s ) +_s 1_s ) = 𝐵)
61	58, 59, 60	sylancl 587	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝐵 -_s 1_s ) +_s 1_s ) = 𝐵)
62	61	breq2d 5111	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑞 <s ((𝐵 -_s 1_s ) +_s 1_s ) ↔ 𝑞 <s 𝐵))
63	53, 57, 62	3bitr3rd 310	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑞 <s 𝐵 ↔ (𝑞 <s (𝐵 -_s 1_s ) ∨ 𝑞 = (𝐵 -_s 1_s ))))
64		simplrl 777	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → 𝑝 ∈ ℕ_0s)
65		simplrr 778	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → 𝑞 ∈ ℕ_0s)
66		peano2n0s 28330	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ_0s → (𝑞 +_s 1_s ) ∈ ℕ_0s)
67	65, 66	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → (𝑞 +_s 1_s ) ∈ ℕ_0s)
68	49	nnn0sd 28328	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝐵 ∈ ℕ_0s)
69		simprl 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑝 ∈ ℕ_0s)
70		n0mulscl 28345	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℕ_0s ∧ 𝑝 ∈ ℕ_0s) → (𝐵 ·_s 𝑝) ∈ ℕ_0s)
71	68, 69, 70	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s 𝑝) ∈ ℕ_0s)
72	71	n0nod 28325	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s 𝑝) ∈ No )
73	59	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 1_s ∈ No )
74	72, 54, 73	addsassd 28006	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s (𝑞 +_s 1_s )))
75	74	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s (𝑞 +_s 1_s )))
76	54, 73, 58	ltaddsubsd 28091	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝑞 +_s 1_s ) <s 𝐵 ↔ 𝑞 <s (𝐵 -_s 1_s )))
77	76	biimpar 477	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → (𝑞 +_s 1_s ) <s 𝐵)
78		oveq2 7368	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑝 → (𝐵 ·_s 𝑟) = (𝐵 ·_s 𝑝))
79	78	oveq1d 7375	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑝 → ((𝐵 ·_s 𝑟) +_s 𝑠) = ((𝐵 ·_s 𝑝) +_s 𝑠))
80	79	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑟 = 𝑝 → ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ↔ (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑠)))
81	80	anbi1d 632	. . . . . . . . . . . . 13 ⊢ (𝑟 = 𝑝 → (((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
82		oveq2 7368	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑞 +_s 1_s ) → ((𝐵 ·_s 𝑝) +_s 𝑠) = ((𝐵 ·_s 𝑝) +_s (𝑞 +_s 1_s )))
83	82	eqeq2d 2748	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑞 +_s 1_s ) → ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑠) ↔ (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s (𝑞 +_s 1_s ))))
84		breq1 5102	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑞 +_s 1_s ) → (𝑠 <s 𝐵 ↔ (𝑞 +_s 1_s ) <s 𝐵))
85	83, 84	anbi12d 633	. . . . . . . . . . . . 13 ⊢ (𝑠 = (𝑞 +_s 1_s ) → (((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s (𝑞 +_s 1_s )) ∧ (𝑞 +_s 1_s ) <s 𝐵)))
86	81, 85	rspc2ev 3590	. . . . . . . . . . . 12 ⊢ ((𝑝 ∈ ℕ_0s ∧ (𝑞 +_s 1_s ) ∈ ℕ_0s ∧ ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s (𝑞 +_s 1_s )) ∧ (𝑞 +_s 1_s ) <s 𝐵)) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))
87	64, 67, 75, 77, 86	syl112anc 1377	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))
88	87	ex 412	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑞 <s (𝐵 -_s 1_s ) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
89		peano2n0s 28330	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ_0s → (𝑝 +_s 1_s ) ∈ ℕ_0s)
90	69, 89	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑝 +_s 1_s ) ∈ ℕ_0s)
91	58	mulsridd 28114	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s 1_s ) = 𝐵)
92	91	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝐵 ·_s 𝑝) +_s (𝐵 ·_s 1_s )) = ((𝐵 ·_s 𝑝) +_s 𝐵))
93	69	n0nod 28325	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑝 ∈ No )
94	58, 93, 73	addsdid 28156	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s (𝑝 +_s 1_s )) = ((𝐵 ·_s 𝑝) +_s (𝐵 ·_s 1_s )))
95	61	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝐵 ·_s 𝑝) +_s ((𝐵 -_s 1_s ) +_s 1_s )) = ((𝐵 ·_s 𝑝) +_s 𝐵))
96	92, 94, 95	3eqtr4rd 2783	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝐵 ·_s 𝑝) +_s ((𝐵 -_s 1_s ) +_s 1_s )) = (𝐵 ·_s (𝑝 +_s 1_s )))
97	72, 55, 73	addsassd 28006	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s ((𝐵 -_s 1_s ) +_s 1_s )))
98		peano2no 27984	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ No → (𝑝 +_s 1_s ) ∈ No )
99	93, 98	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑝 +_s 1_s ) ∈ No )
100	58, 99	mulscld 28135	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s (𝑝 +_s 1_s )) ∈ No )
101	100	addsridd 27965	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s ) = (𝐵 ·_s (𝑝 +_s 1_s )))
102	96, 97, 101	3eqtr4d 2782	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s ))
103	49, 35	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 0_s <s 𝐵)
104		oveq2 7368	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (𝑝 +_s 1_s ) → (𝐵 ·_s 𝑟) = (𝐵 ·_s (𝑝 +_s 1_s )))
105	104	oveq1d 7375	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝑝 +_s 1_s ) → ((𝐵 ·_s 𝑟) +_s 𝑠) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠))
106	105	eqeq2d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑝 +_s 1_s ) → ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ↔ (((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠)))
107	106	anbi1d 632	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑝 +_s 1_s ) → (((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
108		oveq2 7368	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 0_s → ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s ))
109	108	eqeq2d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0_s → ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠) ↔ (((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s )))
110		breq1 5102	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0_s → (𝑠 <s 𝐵 ↔ 0_s <s 𝐵))
111	109, 110	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 0_s → (((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s ) ∧ 0_s <s 𝐵)))
112	107, 111	rspc2ev 3590	. . . . . . . . . . . . 13 ⊢ (((𝑝 +_s 1_s ) ∈ ℕ_0s ∧ 0_s ∈ ℕ_0s ∧ ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s ) ∧ 0_s <s 𝐵)) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))
113	36, 112	mp3an2 1452	. . . . . . . . . . . 12 ⊢ (((𝑝 +_s 1_s ) ∈ ℕ_0s ∧ ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s ) ∧ 0_s <s 𝐵)) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))
114	90, 102, 103, 113	syl12anc 837	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))
115		oveq2 7368	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐵 -_s 1_s ) → ((𝐵 ·_s 𝑝) +_s 𝑞) = ((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )))
116	115	oveq1d 7375	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝐵 -_s 1_s ) → (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = (((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ))
117	116	eqeq1d 2739	. . . . . . . . . . . . 13 ⊢ (𝑞 = (𝐵 -_s 1_s ) → ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ↔ (((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠)))
118	117	anbi1d 632	. . . . . . . . . . . 12 ⊢ (𝑞 = (𝐵 -_s 1_s ) → (((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
119	118	2rexbidv 3202	. . . . . . . . . . 11 ⊢ (𝑞 = (𝐵 -_s 1_s ) → (∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
120	114, 119	syl5ibrcom 247	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑞 = (𝐵 -_s 1_s ) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
121	88, 120	jaod 860	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝑞 <s (𝐵 -_s 1_s ) ∨ 𝑞 = (𝐵 -_s 1_s )) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
122	63, 121	sylbid 240	. . . . . . . 8 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
123		oveq1 7367	. . . . . . . . . . . 12 ⊢ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) → (𝑎 +_s 1_s ) = (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ))
124	123	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) → ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ↔ (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠)))
125	124	anbi1d 632	. . . . . . . . . 10 ⊢ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) → (((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
126	125	2rexbidv 3202	. . . . . . . . 9 ⊢ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) → (∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵) ↔ ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
127	126	imbi2d 340	. . . . . . . 8 ⊢ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) → ((𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)) ↔ (𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))))
128	122, 127	syl5ibrcom 247	. . . . . . 7 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) → (𝑞 <s 𝐵 → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))))
129	128	impd 410	. . . . . 6 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
130	129	rexlimdvva 3194	. . . . 5 ⊢ ((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) → (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵)))
131	130	ex 412	. . . 4 ⊢ (𝑎 ∈ ℕ_0s → (𝐵 ∈ ℕ_s → (∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))))
132	131	a2d 29	. . 3 ⊢ (𝑎 ∈ ℕ_0s → ((𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)) → (𝐵 ∈ ℕ_s → ∃𝑟 ∈ ℕ_0s ∃𝑠 ∈ ℕ_0s ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠) ∧ 𝑠 <s 𝐵))))
133	4, 8, 22, 26, 47, 132	n0sind 28333	. 2 ⊢ (𝐴 ∈ ℕ_0s → (𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
134	133	imp 406	1 ⊢ ((𝐴 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s (𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 ∃wrex 3061 class class class wbr 5099 (class class class)co 7360 No csur 27611 <s clts 27612 ≤s cles 27716 0_s c0s 27805 1_s c1s 27806 +_s cadds 27959 -_s csubs 28020 ·_s cmuls 28106 ℕ_0scn0s 28312 ℕ_scnns 28313

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-nadd 8596 df-no 27614 df-lts 27615 df-bday 27616 df-les 27717 df-slts 27758 df-cuts 27760 df-0s 27807 df-1s 27808 df-made 27827 df-old 27828 df-left 27830 df-right 27831 df-norec 27938 df-norec2 27949 df-adds 27960 df-negs 28021 df-subs 28022 df-muls 28107 df-n0s 28314 df-nns 28315

This theorem is referenced by: z12sge0 28483

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