Theorem halfcut 28472

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 27784	. . . . . . 7 ⊢ (𝜑 → {𝐴} <<s {𝐵})
6	5	cutscld 27797	. . . . . 6 ⊢ (𝜑 → ({𝐴} |s {𝐵}) ∈ No )
7	1, 6	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
8		no2times 28431	. . . . 5 ⊢ (𝐶 ∈ No → (2s ·s 𝐶) = (𝐶 +s 𝐶))
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → (2s ·s 𝐶) = (𝐶 +s 𝐶))
10	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐶 = ({𝐴} |s {𝐵}))
11	5, 5, 10, 10	addsunif 28016	. . . . 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 7367	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 +s 𝐶) = (𝐴 +s 𝐶))
13	12	eqeq2d 2752	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
14	13	rexsng 4611	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
15	2, 14	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
16	15	abbidv 2807	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
17		oveq2 7368	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (𝐶 +s 𝑦) = (𝐶 +s 𝐴))
18	17	eqeq2d 2752	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
19	18	rexsng 4611	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
20	2, 19	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
21	7, 2	addscomd 27981	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 +s 𝐴) = (𝐴 +s 𝐶))
22	21	eqeq2d 2752	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = (𝐶 +s 𝐴) ↔ 𝑥 = (𝐴 +s 𝐶)))
23	20, 22	bitrd 281	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐴 +s 𝐶)))
24	23	abbidv 2807	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
25	16, 24	uneq12d 4102	. . . . . . . 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 4559	. . . . . . . . 9 ⊢ {(𝐴 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}
27		unidm 4090	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}) = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}
28	26, 27	eqtr4i 2767	. . . . . . . 8 ⊢ {(𝐴 +s 𝐶)} = ({𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
29	25, 28	eqtr4di 2794	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) = {(𝐴 +s 𝐶)})
30		oveq1 7367	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐶) = (𝐵 +s 𝐶))
31	30	eqeq2d 2752	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
32	31	rexsng 4611	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
33	3, 32	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
34	33	abbidv 2807	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
35		oveq2 7368	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (𝐶 +s 𝑦) = (𝐶 +s 𝐵))
36	35	eqeq2d 2752	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
37	36	rexsng 4611	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
38	3, 37	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
39	7, 3	addscomd 27981	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 +s 𝐵) = (𝐵 +s 𝐶))
40	39	eqeq2d 2752	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = (𝐶 +s 𝐵) ↔ 𝑥 = (𝐵 +s 𝐶)))
41	38, 40	bitrd 281	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐵 +s 𝐶)))
42	41	abbidv 2807	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
43	34, 42	uneq12d 4102	. . . . . . . 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 4559	. . . . . . . . 9 ⊢ {(𝐵 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}
45		unidm 4090	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}) = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}
46	44, 45	eqtr4i 2767	. . . . . . . 8 ⊢ {(𝐵 +s 𝐶)} = ({𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)} ∪ {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
47	43, 46	eqtr4di 2794	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)}) = {(𝐵 +s 𝐶)})
48	29, 47	oveq12d 7378	. . . . . 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 28433	. . . . . . . . . 10 ⊢ 2s ∈ No
50	49	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2s ∈ No )
51	50, 2	mulscld 28149	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
52	50, 3	mulscld 28149	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐵) ∈ No )
53		2nns 28432	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
54		nnsgt0 28353	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
55	53, 54	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → 0s <s 2s)
56	2, 3, 50, 55	ltmuls2d 28186	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (2s ·s 𝐴) <s (2s ·s 𝐵)))
57	4, 56	mpbid 234	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) <s (2s ·s 𝐵))
58	51, 52, 57	sltssn 27784	. . . . . . 7 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {(2s ·s 𝐵)})
59		no2times 28431	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
60	2, 59	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
61		lesid 27753	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
62	2, 61	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤s 𝐴)
63		breq2 5079	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → (𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
64	63	rexsng 4611	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
65	2, 64	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
66	62, 65	mpbird 259	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥)
67	66	orcd 880	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶))
68		lltr 27876	. . . . . . . . . . . . . 14 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
69	68	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
70		lrcut 27918	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
71	2, 70	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
72	71	eqcomd 2747	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
73	69, 5, 72, 10	ltsrecd 27816	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 <s 𝐶 ↔ (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶)))
74	67, 73	mpbird 259	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 <s 𝐶)
75	2, 7, 74	ltlesd 27759	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ≤s 𝐶)
76	2, 7, 2	leadds2d 28010	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ≤s 𝐶 ↔ (𝐴 +s 𝐴) ≤s (𝐴 +s 𝐶)))
77	75, 76	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐴) ≤s (𝐴 +s 𝐶))
78	60, 77	eqbrtrd 5097	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
79		ovex 7393	. . . . . . . . . 10 ⊢ (2s ·s 𝐴) ∈ V
80		breq1 5078	. . . . . . . . . . 11 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s 𝑦))
81	80	rexbidv 3165	. . . . . . . . . 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 4616	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 +s 𝐶)} (2s ·s 𝐴) ≤s 𝑦)
83		ovex 7393	. . . . . . . . . 10 ⊢ (𝐴 +s 𝐶) ∈ V
84		breq2 5079	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 +s 𝐶) → ((2s ·s 𝐴) ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶)))
85	83, 84	rexsn 4617	. . . . . . . . 9 ⊢ (∃𝑦 ∈ {(𝐴 +s 𝐶)} (2s ·s 𝐴) ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
86	82, 85	bitri 277	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
87	78, 86	sylibr 236	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦)
88		lesid 27753	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ No → 𝐵 ≤s 𝐵)
89	3, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ≤s 𝐵)
90		breq1 5078	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
91	90	rexsng 4611	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
92	3, 91	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
93	89, 92	mpbird 259	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)
94	93	olcd 881	. . . . . . . . . . . 12 ⊢ (𝜑 → (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵))
95		lltr 27876	. . . . . . . . . . . . . 14 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
96	95	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( L ‘𝐵) <<s ( R ‘𝐵))
97		lrcut 27918	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
98	3, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
99	98	eqcomd 2747	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
100	5, 96, 10, 99	ltsrecd 27816	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 <s 𝐵 ↔ (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)))
101	94, 100	mpbird 259	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 <s 𝐵)
102	7, 3, 101	ltlesd 27759	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ≤s 𝐵)
103	7, 3, 3	leadds2d 28010	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ≤s 𝐵 ↔ (𝐵 +s 𝐶) ≤s (𝐵 +s 𝐵)))
104	102, 103	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 +s 𝐶) ≤s (𝐵 +s 𝐵))
105		no2times 28431	. . . . . . . . . 10 ⊢ (𝐵 ∈ No → (2s ·s 𝐵) = (𝐵 +s 𝐵))
106	3, 105	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝐵) = (𝐵 +s 𝐵))
107	104, 106	breqtrrd 5103	. . . . . . . 8 ⊢ (𝜑 → (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
108		ovex 7393	. . . . . . . . . 10 ⊢ (2s ·s 𝐵) ∈ V
109		breq2 5079	. . . . . . . . . . 11 ⊢ (𝑥 = (2s ·s 𝐵) → (𝑦 ≤s 𝑥 ↔ 𝑦 ≤s (2s ·s 𝐵)))
110	109	rexbidv 3165	. . . . . . . . . 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 4616	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ ∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s (2s ·s 𝐵))
112		ovex 7393	. . . . . . . . . 10 ⊢ (𝐵 +s 𝐶) ∈ V
113		breq1 5078	. . . . . . . . . 10 ⊢ (𝑦 = (𝐵 +s 𝐶) → (𝑦 ≤s (2s ·s 𝐵) ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵)))
114	112, 113	rexsn 4617	. . . . . . . . 9 ⊢ (∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s (2s ·s 𝐵) ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
115	111, 114	bitri 277	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
116	107, 115	sylibr 236	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥)
117	2, 7	addscld 27994	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐶) ∈ No )
118	2, 3	addscld 27994	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐵) ∈ No )
119	7, 3, 2	ltadds2d 28011	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐴 +s 𝐵)))
120	101, 119	mpbid 234	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐶) <s (𝐴 +s 𝐵))
121	117, 118, 120	sltssn 27784	. . . . . . . 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 4570	. . . . . . . 8 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} = {(𝐴 +s 𝐵)})
124	121, 123	breqtrrd 5103	. . . . . . 7 ⊢ (𝜑 → {(𝐴 +s 𝐶)} <<s {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})})
125	3, 7	addscld 27994	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 +s 𝐶) ∈ No )
126	3, 2	addscomd 27981	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 +s 𝐴) = (𝐴 +s 𝐵))
127	2, 7, 3	ltadds2d 28011	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 <s 𝐶 ↔ (𝐵 +s 𝐴) <s (𝐵 +s 𝐶)))
128	74, 127	mpbid 234	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 +s 𝐴) <s (𝐵 +s 𝐶))
129	126, 128	eqbrtrrd 5099	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐵) <s (𝐵 +s 𝐶))
130	118, 125, 129	sltssn 27784	. . . . . . . 8 ⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s {(𝐵 +s 𝐶)})
131	123, 130	eqbrtrd 5097	. . . . . . 7 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} <<s {(𝐵 +s 𝐶)})
132	58, 87, 116, 124, 131	cofcut1d 27935	. . . . . 6 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = ({(𝐴 +s 𝐶)} |s {(𝐵 +s 𝐶)}))
133	48, 132, 122	3eqtr2d 2782	. . . . 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 2776	. . . 4 ⊢ (𝜑 → (𝐶 +s 𝐶) = (𝐴 +s 𝐵))
135	9, 134	eqtrd 2776	. . 3 ⊢ (𝜑 → (2s ·s 𝐶) = (𝐴 +s 𝐵))
136		2ne0s 28434	. . . . 5 ⊢ 2s ≠ 0s
137	136	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
138		0no 27823	. . . . . . . . . 10 ⊢ 0s ∈ No
139	138	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s ∈ No )
140		1no 27824	. . . . . . . . . 10 ⊢ 1s ∈ No
141	140	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 1s ∈ No )
142		0lt1s 27826	. . . . . . . . . 10 ⊢ 0s <s 1s
143	142	a1i 11	. . . . . . . . 9 ⊢ (⊤ → 0s <s 1s )
144	139, 141, 143	sltssn 27784	. . . . . . . 8 ⊢ (⊤ → { 0s } <<s { 1s })
145	144	cutscld 27797	. . . . . . 7 ⊢ (⊤ → ({ 0s } |s { 1s }) ∈ No )
146	145	mptru 1555	. . . . . 6 ⊢ ({ 0s } |s { 1s }) ∈ No
147		twocut 28437	. . . . . 6 ⊢ (2s ·s ({ 0s } |s { 1s })) = 1s
148		oveq2 7368	. . . . . . . 8 ⊢ (𝑥 = ({ 0s } |s { 1s }) → (2s ·s 𝑥) = (2s ·s ({ 0s } |s { 1s })))
149	148	eqeq1d 2743	. . . . . . 7 ⊢ (𝑥 = ({ 0s } |s { 1s }) → ((2s ·s 𝑥) = 1s ↔ (2s ·s ({ 0s } |s { 1s })) = 1s ))
150	149	rspcev 3562	. . . . . 6 ⊢ ((({ 0s } |s { 1s }) ∈ No ∧ (2s ·s ({ 0s } |s { 1s })) = 1s ) → ∃𝑥 ∈ No (2s ·s 𝑥) = 1s )
151	146, 147, 150	mp2an 699	. . . . 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 28208	. . 3 ⊢ (𝜑 → (((𝐴 +s 𝐵) /su 2s) = 𝐶 ↔ (2s ·s 𝐶) = (𝐴 +s 𝐵)))
154	135, 153	mpbird 259	. 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) /su 2s) = 𝐶)
155	154	eqcomd 2747	1 ⊢ (𝜑 → 𝐶 = ((𝐴 +s 𝐵) /su 2s))

Syntax hints:   → wi 4   ↔ wb 208   ∨ wo 854   = wceq 1548  ⊤wtru 1549   ∈ wcel 2121  {cab 2719   ≠ wne 2936  ∀wral 3055  ∃wrex 3065   ∪ cun 3883  {csn 4558   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360   No csur 27625   <s clts 27626   ≤s cles 27730   <<s cslts 27771   |s ccuts 27773   0_s c0s 27819   1_s c1s 27820   L cleft 27839   R cright 27840   +_s cadds 27973   ·_s cmuls 28120   /_su cdivs 28201  ℕ_scnns 28327  2_sc2s 28424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  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 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-0s 27821  df-1s 27822  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28035  df-subs 28036  df-muls 28121  df-divs 28202  df-n0s 28328  df-nns 28329  df-2s 28425

This theorem is referenced by:  addhalfcut  28473  pw2cut  28474