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Theorem eucliddivs 28545

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 2765	. . . . . 6 ⊢ (𝑚 = 0_s → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞)))
2	1	anbi1d 642	. . . . 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 3228	. . . 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 343	. . 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 2765	. . . . . 6 ⊢ (𝑚 = 𝑎 → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞)))
6	5	anbi1d 642	. . . . 5 ⊢ (𝑚 = 𝑎 → ((𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
7	6	2rexbidv 3228	. . . 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 343	. . 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 2765	. . . . . . 7 ⊢ (𝑚 = (𝑎 +_s 1_s ) → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ (𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞)))
10	9	anbi1d 642	. . . . . 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 3228	. . . . 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 7418	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → (𝐵 ·_s 𝑝) = (𝐵 ·_s 𝑟))
13	12	oveq1d 7425	. . . . . . . 8 ⊢ (𝑝 = 𝑟 → ((𝐵 ·_s 𝑝) +_s 𝑞) = ((𝐵 ·_s 𝑟) +_s 𝑞))
14	13	eqeq2d 2772	. . . . . . 7 ⊢ (𝑝 = 𝑟 → ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ (𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑞)))
15	14	anbi1d 642	. . . . . 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 7418	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝐵 ·_s 𝑟) +_s 𝑞) = ((𝐵 ·_s 𝑟) +_s 𝑠))
17	16	eqeq2d 2772	. . . . . . 7 ⊢ (𝑞 = 𝑠 → ((𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑞) ↔ (𝑎 +_s 1_s ) = ((𝐵 ·_s 𝑟) +_s 𝑠)))
18		breq1 5111	. . . . . . 7 ⊢ (𝑞 = 𝑠 → (𝑞 <s 𝐵 ↔ 𝑠 <s 𝐵))
19	17, 18	anbi12d 643	. . . . . 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 3246	. . . . 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 290	. . . 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 343	. . 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 2765	. . . . . 6 ⊢ (𝑚 = 𝐴 → (𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞)))
24	23	anbi1d 642	. . . . 5 ⊢ (𝑚 = 𝐴 → ((𝑚 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝐴 = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵)))
25	24	2rexbidv 3228	. . . 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 343	. . 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 28493	. . . . . . 7 ⊢ (𝐵 ∈ ℕ_s → 𝐵 ∈ No )
28		muls01 28281	. . . . . . 7 ⊢ (𝐵 ∈ No → (𝐵 ·_s 0_s ) = 0_s )
29	27, 28	syl 18	. . . . . 6 ⊢ (𝐵 ∈ ℕ_s → (𝐵 ·_s 0_s ) = 0_s )
30	29	oveq1d 7425	. . . . 5 ⊢ (𝐵 ∈ ℕ_s → ((𝐵 ·_s 0_s ) +_s 0_s ) = ( 0_s +_s 0_s ))
31		0no 27978	. . . . . 6 ⊢ 0_s ∈ No
32		addslid 28137	. . . . . 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 2813	. . . 4 ⊢ (𝐵 ∈ ℕ_s → 0_s = ((𝐵 ·_s 0_s ) +_s 0_s ))
35		nnsgt0 28508	. . . 4 ⊢ (𝐵 ∈ ℕ_s → 0_s <s 𝐵)
36		0n0s 28498	. . . . 5 ⊢ 0_s ∈ ℕ_0s
37		oveq2 7418	. . . . . . . . 9 ⊢ (𝑝 = 0_s → (𝐵 ·_s 𝑝) = (𝐵 ·_s 0_s ))
38	37	oveq1d 7425	. . . . . . . 8 ⊢ (𝑝 = 0_s → ((𝐵 ·_s 𝑝) +_s 𝑞) = ((𝐵 ·_s 0_s ) +_s 𝑞))
39	38	eqeq2d 2772	. . . . . . 7 ⊢ (𝑝 = 0_s → ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ↔ 0_s = ((𝐵 ·_s 0_s ) +_s 𝑞)))
40	39	anbi1d 642	. . . . . 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 7418	. . . . . . . 8 ⊢ (𝑞 = 0_s → ((𝐵 ·_s 0_s ) +_s 𝑞) = ((𝐵 ·_s 0_s ) +_s 0_s ))
42	41	eqeq2d 2772	. . . . . . 7 ⊢ (𝑞 = 0_s → ( 0_s = ((𝐵 ·_s 0_s ) +_s 𝑞) ↔ 0_s = ((𝐵 ·_s 0_s ) +_s 0_s )))
43		breq1 5111	. . . . . . 7 ⊢ (𝑞 = 0_s → (𝑞 <s 𝐵 ↔ 0_s <s 𝐵))
44	42, 43	anbi12d 643	. . . . . 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 3593	. . . . 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 1478	. . . 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 595	. . 3 ⊢ (𝐵 ∈ ℕ_s → ∃𝑝 ∈ ℕ_0s ∃𝑞 ∈ ℕ_0s ( 0_s = ((𝐵 ·_s 𝑝) +_s 𝑞) ∧ 𝑞 <s 𝐵))
48		simprr 784	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑞 ∈ ℕ_0s)
49		simplr 780	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝐵 ∈ ℕ_s)
50		nnm1n0s 28544	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕ_s → (𝐵 -_s 1_s ) ∈ ℕ_0s)
51	49, 50	syl 18	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 -_s 1_s ) ∈ ℕ_0s)
52		n0lesltp1 28535	. . . . . . . . . . 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 595	. . . . . . . . . 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 28494	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑞 ∈ No )
55	51	n0nod 28494	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 -_s 1_s ) ∈ No )
56		lesloe 27894	. . . . . . . . . . 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 595	. . . . . . . . . 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 28495	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝐵 ∈ No )
59		1no 27979	. . . . . . . . . . . 12 ⊢ 1_s ∈ No
60		npcans 28244	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 1_s ∈ No ) → ((𝐵 -_s 1_s ) +_s 1_s ) = 𝐵)
61	58, 59, 60	sylancl 597	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → ((𝐵 -_s 1_s ) +_s 1_s ) = 𝐵)
62	61	breq2d 5120	. . . . . . . . . 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 313	. . . . . . . . 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 788	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → 𝑝 ∈ ℕ_0s)
65		simplrr 789	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) ∧ 𝑞 <s (𝐵 -_s 1_s )) → 𝑞 ∈ ℕ_0s)
66		peano2n0s 28499	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ_0s → (𝑞 +_s 1_s ) ∈ ℕ_0s)
67	65, 66	syl 18	. . . . . . . . . . . 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 28497	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝐵 ∈ ℕ_0s)
69		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑝 ∈ ℕ_0s)
70		n0mulscl 28514	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℕ_0s ∧ 𝑝 ∈ ℕ_0s) → (𝐵 ·_s 𝑝) ∈ ℕ_0s)
71	68, 69, 70	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s 𝑝) ∈ ℕ_0s)
72	71	n0nod 28494	. . . . . . . . . . . . . 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 28175	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 28260	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑝 → (𝐵 ·_s 𝑟) = (𝐵 ·_s 𝑝))
79	78	oveq1d 7425	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑝 → ((𝐵 ·_s 𝑟) +_s 𝑠) = ((𝐵 ·_s 𝑝) +_s 𝑠))
80	79	eqeq2d 2772	. . . . . . . . . . . . . 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 642	. . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑞 +_s 1_s ) → ((𝐵 ·_s 𝑝) +_s 𝑠) = ((𝐵 ·_s 𝑝) +_s (𝑞 +_s 1_s )))
83	82	eqeq2d 2772	. . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑞 +_s 1_s ) → (𝑠 <s 𝐵 ↔ (𝑞 +_s 1_s ) <s 𝐵))
85	83, 84	anbi12d 643	. . . . . . . . . . . . 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 3593	. . . . . . . . . . . 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 1399	. . . . . . . . . . 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 417	. . . . . . . . . 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 28499	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ_0s → (𝑝 +_s 1_s ) ∈ ℕ_0s)
90	69, 89	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑝 +_s 1_s ) ∈ ℕ_0s)
91	58	mulsridd 28283	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s 1_s ) = 𝐵)
92	91	oveq2d 7426	. . . . . . . . . . . . . 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 28494	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 𝑝 ∈ No )
94	58, 93, 73	addsdid 28325	. . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . 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 28175	. . . . . . . . . . . . 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 28153	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ No → (𝑝 +_s 1_s ) ∈ No )
99	93, 98	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝑝 +_s 1_s ) ∈ No )
100	58, 99	mulscld 28304	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → (𝐵 ·_s (𝑝 +_s 1_s )) ∈ No )
101	100	addsridd 28134	. . . . . . . . . . . . 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 2806	. . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ_0s ∧ 𝐵 ∈ ℕ_s) ∧ (𝑝 ∈ ℕ_0s ∧ 𝑞 ∈ ℕ_0s)) → 0_s <s 𝐵)
104		oveq2 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (𝑝 +_s 1_s ) → (𝐵 ·_s 𝑟) = (𝐵 ·_s (𝑝 +_s 1_s )))
105	104	oveq1d 7425	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝑝 +_s 1_s ) → ((𝐵 ·_s 𝑟) +_s 𝑠) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠))
106	105	eqeq2d 2772	. . . . . . . . . . . . . . 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 642	. . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 0_s → ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 𝑠) = ((𝐵 ·_s (𝑝 +_s 1_s )) +_s 0_s ))
109	108	eqeq2d 2772	. . . . . . . . . . . . . . 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 5111	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0_s → (𝑠 <s 𝐵 ↔ 0_s <s 𝐵))
111	109, 110	anbi12d 643	. . . . . . . . . . . . . 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 3593	. . . . . . . . . . . . 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 1476	. . . . . . . . . . . 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 849	. . . . . . . . . . 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 7418	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐵 -_s 1_s ) → ((𝐵 ·_s 𝑝) +_s 𝑞) = ((𝐵 ·_s 𝑝) +_s (𝐵 -_s 1_s )))
116	115	oveq1d 7425	. . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . 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 642	. . . . . . . . . . . 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 3228	. . . . . . . . . . 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 250	. . . . . . . . . 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 872	. . . . . . . . 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 243	. . . . . . . 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 7417	. . . . . . . . . . . 12 ⊢ (𝑎 = ((𝐵 ·_s 𝑝) +_s 𝑞) → (𝑎 +_s 1_s ) = (((𝐵 ·_s 𝑝) +_s 𝑞) +_s 1_s ))
124	123	eqeq1d 2763	. . . . . . . . . . 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 642	. . . . . . . . . 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 3228	. . . . . . . . 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 343	. . . . . . . 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 250	. . . . . . 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 415	. . . . . 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 3220	. . . . 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 417	. . . 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 30	. . 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 28502	. 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 411	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 209 ∧ wa 400 ∨ wo 860 = wceq 1568 ∈ wcel 2141 ∃wrex 3087 class class class wbr 5108 (class class class)co 7410 No csur 27780 <s clts 27781 ≤s cles 27884 0_s c0s 27974 1_s c1s 27975 +_s cadds 28128 -_s csubs 28189 ·_s cmuls 28275 ℕ_0scn0s 28481 ℕ_scnns 28482

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-ot 4597 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-nadd 8651 df-no 27783 df-lts 27784 df-bday 27785 df-les 27885 df-slts 27927 df-cuts 27929 df-0s 27976 df-1s 27977 df-made 27996 df-old 27997 df-left 27999 df-right 28000 df-norec 28107 df-norec2 28118 df-adds 28129 df-negs 28190 df-subs 28191 df-muls 28276 df-n0s 28483 df-nns 28484

This theorem is referenced by: z12sge0 28652

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