Theorem halfcut 28617

Description: Relate the cut of twice of two numbers to the cut of the numbers. Lemma 4.2 of [Gonshor] p. 28. (Contributed by Scott Fenton, 7-Aug-2025.) Avoid the axiom of infinity. (Proof modified by Scott Fenton, 6-Sep-2025.)

Hypotheses

Ref	Expression
halfcut.1	⊢ (𝜑 → 𝐴 ∈ No )
halfcut.2	⊢ (𝜑 → 𝐵 ∈ No )
halfcut.3	⊢ (𝜑 → 𝐴 <s 𝐵)
halfcut.4	⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))
halfcut.5	⊢ 𝐶 = ({𝐴} |s {𝐵})

Assertion

Ref	Expression
halfcut	⊢ (𝜑 → 𝐶 = ((𝐴 +s 𝐵) /su 2s))

Proof of Theorem halfcut

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		halfcut.5	. . . . . 6 ⊢ 𝐶 = ({𝐴} |s {𝐵})
2		halfcut.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ No )
3		halfcut.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ No )
4		halfcut.3	. . . . . . . 8 ⊢ (𝜑 → 𝐴 <s 𝐵)
5	2, 3, 4	sltssn 27929	. . . . . . 7 ⊢ (𝜑 → {𝐴} <<s {𝐵})
6	5	cutscld 27942	. . . . . 6 ⊢ (𝜑 → ({𝐴} |s {𝐵}) ∈ No )
7	1, 6	eqeltrid 2873	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
8		no2times 28576	. . . . 5 ⊢ (𝐶 ∈ No → (2s ·s 𝐶) = (𝐶 +s 𝐶))
9	7, 8	syl 18	. . . 4 ⊢ (𝜑 → (2s ·s 𝐶) = (𝐶 +s 𝐶))
10	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐶 = ({𝐴} |s {𝐵}))
11	5, 5, 10, 10	addsunif 28161	. . . . 5 ⊢ (𝜑 → (𝐶 +s 𝐶) = (({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)})))
12		oveq1 7418	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 +s 𝐶) = (𝐴 +s 𝐶))
13	12	eqeq2d 2780	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
14	13	rexsng 4647	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
15	2, 14	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
16	15	abbidv 2835	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
17		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (𝐶 +s 𝑦) = (𝐶 +s 𝐴))
18	17	eqeq2d 2780	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
19	18	rexsng 4647	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
20	2, 19	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
21	7, 2	addscomd 28126	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 +s 𝐴) = (𝐴 +s 𝐶))
22	21	eqeq2d 2780	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = (𝐶 +s 𝐴) ↔ 𝑥 = (𝐴 +s 𝐶)))
23	20, 22	bitrd 282	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐴 +s 𝐶)))
24	23	abbidv 2835	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
25	16, 24	uneq12d 4131	. . . . . . . 8 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) = ({𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}))
26		df-sn 4595	. . . . . . . . 9 ⊢ {(𝐴 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}
27		unidm 4119	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}) = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}
28	26, 27	eqtr4i 2795	. . . . . . . 8 ⊢ {(𝐴 +s 𝐶)} = ({𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
29	25, 28	eqtr4di 2822	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) = {(𝐴 +s 𝐶)})
30		oveq1 7418	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐶) = (𝐵 +s 𝐶))
31	30	eqeq2d 2780	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
32	31	rexsng 4647	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
33	3, 32	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
34	33	abbidv 2835	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
35		oveq2 7419	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (𝐶 +s 𝑦) = (𝐶 +s 𝐵))
36	35	eqeq2d 2780	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
37	36	rexsng 4647	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
38	3, 37	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
39	7, 3	addscomd 28126	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 +s 𝐵) = (𝐵 +s 𝐶))
40	39	eqeq2d 2780	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = (𝐶 +s 𝐵) ↔ 𝑥 = (𝐵 +s 𝐶)))
41	38, 40	bitrd 282	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐵 +s 𝐶)))
42	41	abbidv 2835	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
43	34, 42	uneq12d 4131	. . . . . . . 8 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)}) = ({𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}))
44		df-sn 4595	. . . . . . . . 9 ⊢ {(𝐵 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}
45		unidm 4119	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}) = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}
46	44, 45	eqtr4i 2795	. . . . . . . 8 ⊢ {(𝐵 +s 𝐶)} = ({𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
47	43, 46	eqtr4di 2822	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)}) = {(𝐵 +s 𝐶)})
48	29, 47	oveq12d 7429	. . . . . 6 ⊢ (𝜑 → (({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)})) = ({(𝐴 +s 𝐶)} |s {(𝐵 +s 𝐶)}))
49		2no 28578	. . . . . . . . . 10 ⊢ 2s ∈ No
50	49	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2s ∈ No )
51	50, 2	mulscld 28294	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
52	50, 3	mulscld 28294	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐵) ∈ No )
53		2nns 28577	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
54		nnsgt0 28498	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
55	53, 54	mp1i 14	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s 2s)
56	2, 3, 50, 55	ltmuls2d 28331	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (2s ·s 𝐴) <s (2s ·s 𝐵)))
57	4, 56	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) <s (2s ·s 𝐵))
58	51, 52, 57	sltssn 27929	. . . . . . 7 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {(2s ·s 𝐵)})
59		no2times 28576	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
60	2, 59	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
61		lesid 27897	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
62	2, 61	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤s 𝐴)
63		breq2 5117	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → (𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
64	63	rexsng 4647	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
65	2, 64	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
66	62, 65	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥)
67	66	orcd 886	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶))
68		lltr 28021	. . . . . . . . . . . . . 14 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
69	68	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
70		lrcut 28063	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
71	2, 70	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
72	71	eqcomd 2775	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
73	69, 5, 72, 10	ltsrecd 27961	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 <s 𝐶 ↔ (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶)))
74	67, 73	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 <s 𝐶)
75	2, 7, 74	ltlesd 27903	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤s 𝐶)
76	2, 7, 2	leadds2d 28155	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ≤s 𝐶 ↔ (𝐴 +s 𝐴) ≤s (𝐴 +s 𝐶)))
77	75, 76	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐴) ≤s (𝐴 +s 𝐶))
78	60, 77	eqbrtrd 5137	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
79		ovex 7444	. . . . . . . . . 10 ⊢ (2s ·s 𝐴) ∈ V
80		breq1 5116	. . . . . . . . . . 11 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s 𝑦))
81	80	rexbidv 3195	. . . . . . . . . 10 ⊢ (𝑥 = (2s ·s 𝐴) → (∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 +s 𝐶)} (2s ·s 𝐴) ≤s 𝑦))
82	79, 81	ralsn 4652	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 +s 𝐶)} (2s ·s 𝐴) ≤s 𝑦)
83		ovex 7444	. . . . . . . . . 10 ⊢ (𝐴 +s 𝐶) ∈ V
84		breq2 5117	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 +s 𝐶) → ((2s ·s 𝐴) ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶)))
85	83, 84	rexsn 4653	. . . . . . . . 9 ⊢ (∃𝑦 ∈ {(𝐴 +s 𝐶)} (2s ·s 𝐴) ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
86	82, 85	bitri 278	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
87	78, 86	sylibr 237	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦)
88		lesid 27897	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ No → 𝐵 ≤s 𝐵)
89	3, 88	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ≤s 𝐵)
90		breq1 5116	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
91	90	rexsng 4647	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
92	3, 91	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
93	89, 92	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)
94	93	olcd 887	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵))
95		lltr 28021	. . . . . . . . . . . . . 14 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
96	95	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( L ‘𝐵) <<s ( R ‘𝐵))
97		lrcut 28063	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
98	3, 97	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
99	98	eqcomd 2775	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
100	5, 96, 10, 99	ltsrecd 27961	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 <s 𝐵 ↔ (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)))
101	94, 100	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 <s 𝐵)
102	7, 3, 101	ltlesd 27903	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≤s 𝐵)
103	7, 3, 3	leadds2d 28155	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ≤s 𝐵 ↔ (𝐵 +s 𝐶) ≤s (𝐵 +s 𝐵)))
104	102, 103	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 +s 𝐶) ≤s (𝐵 +s 𝐵))
105		no2times 28576	. . . . . . . . . 10 ⊢ (𝐵 ∈ No → (2s ·s 𝐵) = (𝐵 +s 𝐵))
106	3, 105	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝐵) = (𝐵 +s 𝐵))
107	104, 106	breqtrrd 5143	. . . . . . . 8 ⊢ (𝜑 → (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
108		ovex 7444	. . . . . . . . . 10 ⊢ (2s ·s 𝐵) ∈ V
109		breq2 5117	. . . . . . . . . . 11 ⊢ (𝑥 = (2s ·s 𝐵) → (𝑦 ≤s 𝑥 ↔ 𝑦 ≤s (2s ·s 𝐵)))
110	109	rexbidv 3195	. . . . . . . . . 10 ⊢ (𝑥 = (2s ·s 𝐵) → (∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ ∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s (2s ·s 𝐵)))
111	108, 110	ralsn 4652	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ ∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s (2s ·s 𝐵))
112		ovex 7444	. . . . . . . . . 10 ⊢ (𝐵 +s 𝐶) ∈ V
113		breq1 5116	. . . . . . . . . 10 ⊢ (𝑦 = (𝐵 +s 𝐶) → (𝑦 ≤s (2s ·s 𝐵) ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵)))
114	112, 113	rexsn 4653	. . . . . . . . 9 ⊢ (∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s (2s ·s 𝐵) ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
115	111, 114	bitri 278	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
116	107, 115	sylibr 237	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥)
117	2, 7	addscld 28139	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐶) ∈ No )
118	2, 3	addscld 28139	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐵) ∈ No )
119	7, 3, 2	ltadds2d 28156	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐴 +s 𝐵)))
120	101, 119	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐶) <s (𝐴 +s 𝐵))
121	117, 118, 120	sltssn 27929	. . . . . . . 8 ⊢ (𝜑 → {(𝐴 +s 𝐶)} <<s {(𝐴 +s 𝐵)})
122		halfcut.4	. . . . . . . . 9 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))
123	122	sneqd 4606	. . . . . . . 8 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} = {(𝐴 +s 𝐵)})
124	121, 123	breqtrrd 5143	. . . . . . 7 ⊢ (𝜑 → {(𝐴 +s 𝐶)} <<s {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})})
125	3, 7	addscld 28139	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 +s 𝐶) ∈ No )
126	3, 2	addscomd 28126	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 +s 𝐴) = (𝐴 +s 𝐵))
127	2, 7, 3	ltadds2d 28156	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 <s 𝐶 ↔ (𝐵 +s 𝐴) <s (𝐵 +s 𝐶)))
128	74, 127	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 +s 𝐴) <s (𝐵 +s 𝐶))
129	126, 128	eqbrtrrd 5139	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐵) <s (𝐵 +s 𝐶))
130	118, 125, 129	sltssn 27929	. . . . . . . 8 ⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s {(𝐵 +s 𝐶)})
131	123, 130	eqbrtrd 5137	. . . . . . 7 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} <<s {(𝐵 +s 𝐶)})
132	58, 87, 116, 124, 131	cofcut1d 28080	. . . . . 6 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = ({(𝐴 +s 𝐶)} |s {(𝐵 +s 𝐶)}))
133	48, 132, 122	3eqtr2d 2810	. . . . 5 ⊢ (𝜑 → (({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)})) = (𝐴 +s 𝐵))
134	11, 133	eqtrd 2804	. . . 4 ⊢ (𝜑 → (𝐶 +s 𝐶) = (𝐴 +s 𝐵))
135	9, 134	eqtrd 2804	. . 3 ⊢ (𝜑 → (2s ·s 𝐶) = (𝐴 +s 𝐵))
136		2ne0s 28579	. . . . 5 ⊢ 2s ≠ 0s
137	136	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
138		0no 27968	. . . . . . . . . 10 ⊢ 0s ∈ No
139	138	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s ∈ No )
140		1no 27969	. . . . . . . . . 10 ⊢ 1s ∈ No
141	140	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 1s ∈ No )
142		0lt1s 27971	. . . . . . . . . 10 ⊢ 0s <s 1s
143	142	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s <s 1s )
144	139, 141, 143	sltssn 27929	. . . . . . . 8 ⊢ (⊤ → { 0s } <<s { 1s })
145	144	cutscld 27942	. . . . . . 7 ⊢ (⊤ → ({ 0s } |s { 1s }) ∈ No )
146	145	mptru 1574	. . . . . 6 ⊢ ({ 0s } |s { 1s }) ∈ No
147		twocut 28582	. . . . . 6 ⊢ (2s ·s ({ 0s } |s { 1s })) = 1s
148		oveq2 7419	. . . . . . . 8 ⊢ (𝑥 = ({ 0s } |s { 1s }) → (2s ·s 𝑥) = (2s ·s ({ 0s } |s { 1s })))
149	148	eqeq1d 2771	. . . . . . 7 ⊢ (𝑥 = ({ 0s } |s { 1s }) → ((2s ·s 𝑥) = 1s ↔ (2s ·s ({ 0s } |s { 1s })) = 1s ))
150	149	rspcev 3590	. . . . . 6 ⊢ ((({ 0s } |s { 1s }) ∈ No ∧ (2s ·s ({ 0s } |s { 1s })) = 1s ) → ∃𝑥 ∈ No (2s ·s 𝑥) = 1s )
151	146, 147, 150	mp2an 704	. . . . 5 ⊢ ∃𝑥 ∈ No (2s ·s 𝑥) = 1s
152	151	a1i 11	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ No (2s ·s 𝑥) = 1s )
153	118, 7, 50, 137, 152	divmulswd 28353	. . 3 ⊢ (𝜑 → (((𝐴 +s 𝐵) /su 2s) = 𝐶 ↔ (2s ·s 𝐶) = (𝐴 +s 𝐵)))
154	135, 153	mpbird 260	. 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) /su 2s) = 𝐶)
155	154	eqcomd 2775	1 ⊢ (𝜑 → 𝐶 = ((𝐴 +s 𝐵) /su 2s))

Syntax hints:   → wi 4   ↔ wb 209   ∨ wo 860   = wceq 1567  ⊤wtru 1568   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∪ cun 3911  {csn 4594   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411   No csur 27770   <s clts 27771   ≤s cles 27874   <<s cslts 27916   |s ccuts 27918   0_s c0s 27964   1_s c1s 27965   L cleft 27984   R cright 27985   +_s cadds 28118   ·_s cmuls 28265   /_su cdivs 28346  ℕ_scnns 28472  2_sc2s 28569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-nadd 8652  df-no 27773  df-lts 27774  df-bday 27775  df-les 27875  df-slts 27917  df-cuts 27919  df-0s 27966  df-1s 27967  df-made 27986  df-old 27987  df-left 27989  df-right 27990  df-norec 28097  df-norec2 28108  df-adds 28119  df-negs 28180  df-subs 28181  df-muls 28266  df-divs 28347  df-n0s 28473  df-nns 28474  df-2s 28570

This theorem is referenced by:  addhalfcut  28618  pw2cut  28619