Theorem eucliddivs 28762

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 2764	. . . . . 6 ⊢ (𝑚 = 0s → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 0s = ((𝐵 ·s 𝑝) +s 𝑞)))
2	1	anbi1d 643	. . . . 5 ⊢ (𝑚 = 0s → ((𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
3	2	2rexbidv 3227	. . . 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 343	. . 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 2764	. . . . . 6 ⊢ (𝑚 = 𝑎 → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 𝑎 = ((𝐵 ·s 𝑝) +s 𝑞)))
6	5	anbi1d 643	. . . . 5 ⊢ (𝑚 = 𝑎 → ((𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
7	6	2rexbidv 3227	. . . 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 343	. . 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 2764	. . . . . . 7 ⊢ (𝑚 = (𝑎 +s 1s ) → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ (𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞)))
10	9	anbi1d 643	. . . . . 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 3227	. . . . 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 7428	. . . . . . . . 9 ⊢ (𝑝 = 𝑟 → (𝐵 ·s 𝑝) = (𝐵 ·s 𝑟))
13	12	oveq1d 7435	. . . . . . . 8 ⊢ (𝑝 = 𝑟 → ((𝐵 ·s 𝑝) +s 𝑞) = ((𝐵 ·s 𝑟) +s 𝑞))
14	13	eqeq2d 2771	. . . . . . 7 ⊢ (𝑝 = 𝑟 → ((𝑎 +s 1s ) = ((𝐵 ·s 𝑝) +s 𝑞) ↔ (𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑞)))
15	14	anbi1d 643	. . . . . 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 7428	. . . . . . . 8 ⊢ (𝑞 = 𝑠 → ((𝐵 ·s 𝑟) +s 𝑞) = ((𝐵 ·s 𝑟) +s 𝑠))
17	16	eqeq2d 2771	. . . . . . 7 ⊢ (𝑞 = 𝑠 → ((𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑞) ↔ (𝑎 +s 1s ) = ((𝐵 ·s 𝑟) +s 𝑠)))
18		breq1 5105	. . . . . . 7 ⊢ (𝑞 = 𝑠 → (𝑞 <s 𝐵 ↔ 𝑠 <s 𝐵))
19	17, 18	anbi12d 644	. . . . . 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 3245	. . . . 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 290	. . . 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 343	. . 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 2764	. . . . . 6 ⊢ (𝑚 = 𝐴 → (𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 𝐴 = ((𝐵 ·s 𝑝) +s 𝑞)))
24	23	anbi1d 643	. . . . 5 ⊢ (𝑚 = 𝐴 → ((𝑚 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
25	24	2rexbidv 3227	. . . 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 343	. . 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 28710	. . . . . . 7 ⊢ (𝐵 ∈ ℕs → 𝐵 ∈ No)
28		muls01 28498	. . . . . . 7 ⊢ (𝐵 ∈ No → (𝐵 ·s 0s ) = 0s )
29	27, 28	syl 18	. . . . . 6 ⊢ (𝐵 ∈ ℕs → (𝐵 ·s 0s ) = 0s )
30	29	oveq1d 7435	. . . . 5 ⊢ (𝐵 ∈ ℕs → ((𝐵 ·s 0s ) +s 0s ) = ( 0s +s 0s ))
31		0no 28195	. . . . . 6 ⊢ 0s ∈ No
32		addslid 28354	. . . . . 6 ⊢ ( 0s ∈ No → ( 0s +s 0s ) = 0s )
33	31, 32	ax-mp 5	. . . . 5 ⊢ ( 0s +s 0s ) = 0s
34	30, 33	eqtr2di 2812	. . . 4 ⊢ (𝐵 ∈ ℕs → 0s = ((𝐵 ·s 0s ) +s 0s ))
35		nnsgt0 28725	. . . 4 ⊢ (𝐵 ∈ ℕs → 0s <s 𝐵)
36		0n0s 28715	. . . . 5 ⊢ 0s ∈ ℕ0s
37		oveq2 7428	. . . . . . . . 9 ⊢ (𝑝 = 0s → (𝐵 ·s 𝑝) = (𝐵 ·s 0s ))
38	37	oveq1d 7435	. . . . . . . 8 ⊢ (𝑝 = 0s → ((𝐵 ·s 𝑝) +s 𝑞) = ((𝐵 ·s 0s ) +s 𝑞))
39	38	eqeq2d 2771	. . . . . . 7 ⊢ (𝑝 = 0s → ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ↔ 0s = ((𝐵 ·s 0s ) +s 𝑞)))
40	39	anbi1d 643	. . . . . 6 ⊢ (𝑝 = 0s → (( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵) ↔ ( 0s = ((𝐵 ·s 0s ) +s 𝑞) ∧ 𝑞 <s 𝐵)))
41		oveq2 7428	. . . . . . . 8 ⊢ (𝑞 = 0s → ((𝐵 ·s 0s ) +s 𝑞) = ((𝐵 ·s 0s ) +s 0s ))
42	41	eqeq2d 2771	. . . . . . 7 ⊢ (𝑞 = 0s → ( 0s = ((𝐵 ·s 0s ) +s 𝑞) ↔ 0s = ((𝐵 ·s 0s ) +s 0s )))
43		breq1 5105	. . . . . . 7 ⊢ (𝑞 = 0s → (𝑞 <s 𝐵 ↔ 0s <s 𝐵))
44	42, 43	anbi12d 644	. . . . . 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 3588	. . . . 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 1480	. . . 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 596	. . 3 ⊢ (𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s ( 0s = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))
48		simprr 785	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑞 ∈ ℕ0s)
49		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝐵 ∈ ℕs)
50		nnm1n0s 28761	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℕs → (𝐵 −s 1s ) ∈ ℕ0s)
51	49, 50	syl 18	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 −s 1s ) ∈ ℕ0s)
52		n0lesltp1 28752	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℕ0s ∧ (𝐵 −s 1s ) ∈ ℕ0s) → (𝑞 ≤s (𝐵 −s 1s ) ↔ 𝑞 <s ((𝐵 −s 1s ) +s 1s )))
53	48, 51, 52	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 ≤s (𝐵 −s 1s ) ↔ 𝑞 <s ((𝐵 −s 1s ) +s 1s )))
54	48	n0nod 28711	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑞 ∈ No)
55	51	n0nod 28711	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 −s 1s ) ∈ No)
56		lesloe 28111	. . . . . . . . . . 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 596	. . . . . . . . . 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 28712	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝐵 ∈ No)
59		1no 28196	. . . . . . . . . . . 12 ⊢ 1s ∈ No
60		npcans 28461	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ No ∧ 1s ∈ No) → ((𝐵 −s 1s ) +s 1s ) = 𝐵)
61	58, 59, 60	sylancl 598	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝐵 −s 1s ) +s 1s ) = 𝐵)
62	61	breq2d 5114	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 <s ((𝐵 −s 1s ) +s 1s ) ↔ 𝑞 <s 𝐵))
63	53, 57, 62	3bitr3rd 313	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑞 <s 𝐵 ↔ (𝑞 <s (𝐵 −s 1s ) ∨ 𝑞 = (𝐵 −s 1s ))))
64		simplrl 789	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 −s 1s )) → 𝑝 ∈ ℕ0s)
65		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 −s 1s )) → 𝑞 ∈ ℕ0s)
66		peano2n0s 28716	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ ℕ0s → (𝑞 +s 1s ) ∈ ℕ0s)
67	65, 66	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 −s 1s )) → (𝑞 +s 1s ) ∈ ℕ0s)
68	49	nnn0sd 28714	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝐵 ∈ ℕ0s)
69		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
70		n0mulscl 28731	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (𝐵 ·s 𝑝) ∈ ℕ0s)
71	68, 69, 70	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s 𝑝) ∈ ℕ0s)
72	71	n0nod 28711	. . . . . . . . . . . . . 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 28392	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 28477	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝑞 +s 1s ) <s 𝐵 ↔ 𝑞 <s (𝐵 −s 1s )))
77	76	biimpar 483	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) ∧ 𝑞 <s (𝐵 −s 1s )) → (𝑞 +s 1s ) <s 𝐵)
78		oveq2 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = 𝑝 → (𝐵 ·s 𝑟) = (𝐵 ·s 𝑝))
79	78	oveq1d 7435	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = 𝑝 → ((𝐵 ·s 𝑟) +s 𝑠) = ((𝐵 ·s 𝑝) +s 𝑠))
80	79	eqeq2d 2771	. . . . . . . . . . . . . 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 643	. . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = (𝑞 +s 1s ) → ((𝐵 ·s 𝑝) +s 𝑠) = ((𝐵 ·s 𝑝) +s (𝑞 +s 1s )))
83	82	eqeq2d 2771	. . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . 14 ⊢ (𝑠 = (𝑞 +s 1s ) → (𝑠 <s 𝐵 ↔ (𝑞 +s 1s ) <s 𝐵))
85	83, 84	anbi12d 644	. . . . . . . . . . . . 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 3588	. . . . . . . . . . . 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 1401	. . . . . . . . . . 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 418	. . . . . . . . . 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 28716	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ ℕ0s → (𝑝 +s 1s ) ∈ ℕ0s)
90	69, 89	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑝 +s 1s ) ∈ ℕ0s)
91	58	mulsridd 28500	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s 1s ) = 𝐵)
92	91	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → ((𝐵 ·s 𝑝) +s (𝐵 ·s 1s )) = ((𝐵 ·s 𝑝) +s 𝐵))
93	69	n0nod 28711	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 𝑝 ∈ No)
94	58, 93, 73	addsdid 28542	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s (𝑝 +s 1s )) = ((𝐵 ·s 𝑝) +s (𝐵 ·s 1s )))
95	61	oveq2d 7436	. . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . 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 28392	. . . . . . . . . . . . 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 28370	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ No → (𝑝 +s 1s ) ∈ No)
99	93, 98	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝑝 +s 1s ) ∈ No)
100	58, 99	mulscld 28521	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → (𝐵 ·s (𝑝 +s 1s )) ∈ No)
101	100	addsridd 28351	. . . . . . . . . . . . 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 2805	. . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) ∧ (𝑝 ∈ ℕ0s ∧ 𝑞 ∈ ℕ0s)) → 0s <s 𝐵)
104		oveq2 7428	. . . . . . . . . . . . . . . . 17 ⊢ (𝑟 = (𝑝 +s 1s ) → (𝐵 ·s 𝑟) = (𝐵 ·s (𝑝 +s 1s )))
105	104	oveq1d 7435	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝑝 +s 1s ) → ((𝐵 ·s 𝑟) +s 𝑠) = ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠))
106	105	eqeq2d 2771	. . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 0s → ((𝐵 ·s (𝑝 +s 1s )) +s 𝑠) = ((𝐵 ·s (𝑝 +s 1s )) +s 0s ))
109	108	eqeq2d 2771	. . . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . 15 ⊢ (𝑠 = 0s → (𝑠 <s 𝐵 ↔ 0s <s 𝐵))
111	109, 110	anbi12d 644	. . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . 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 1478	. . . . . . . . . . . 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 850	. . . . . . . . . . 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 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = (𝐵 −s 1s ) → ((𝐵 ·s 𝑝) +s 𝑞) = ((𝐵 ·s 𝑝) +s (𝐵 −s 1s )))
116	115	oveq1d 7435	. . . . . . . . . . . . . 14 ⊢ (𝑞 = (𝐵 −s 1s ) → (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ) = (((𝐵 ·s 𝑝) +s (𝐵 −s 1s )) +s 1s ))
117	116	eqeq1d 2762	. . . . . . . . . . . . 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 643	. . . . . . . . . . . 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 3227	. . . . . . . . . . 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 250	. . . . . . . . . 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 873	. . . . . . . . 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 243	. . . . . . . 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 7427	. . . . . . . . . . . 12 ⊢ (𝑎 = ((𝐵 ·s 𝑝) +s 𝑞) → (𝑎 +s 1s ) = (((𝐵 ·s 𝑝) +s 𝑞) +s 1s ))
124	123	eqeq1d 2762	. . . . . . . . . . 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 643	. . . . . . . . . 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 3227	. . . . . . . . 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 343	. . . . . . . 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 250	. . . . . . 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 416	. . . . . 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 3219	. . . . 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 418	. . . 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 30	. . 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 28719	. 2 ⊢ (𝐴 ∈ ℕ0s → (𝐵 ∈ ℕs → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵)))
134	133	imp 412	1 ⊢ ((𝐴 ∈ ℕ0s ∧ 𝐵 ∈ ℕs) → ∃𝑝 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝐴 = ((𝐵 ·s 𝑝) +s 𝑞) ∧ 𝑞 <s 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  (class class class)co 7420  Nocsur 27997   <s clts 27998   ≤s cles 28101   0_s c0s 28191   1_s c1s 28192   +_s cadds 28345   −_s csubs 28406   ·_s cmuls 28492  ℕ_0scn0s 28698  ℕ_scnns 28699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493  df-n0s 28700  df-nns 28701

This theorem is used by:  z12sge0  28869