Theorem halfcut 28416

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.)

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	ssltsn 27837	. . . . . . 7 ⊢ (𝜑 → {𝐴} <<s {𝐵})
6	5	scutcld 27848	. . . . . 6 ⊢ (𝜑 → ({𝐴} |s {𝐵}) ∈ No )
7	1, 6	eqeltrid 2845	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ No )
8		no2times 28401	. . . . 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	. . . . 5 ⊢ (𝜑 → 𝐶 = ({𝐴} |s {𝐵}))
11	5, 5, 10, 10	addsunif 28035	. . . 4 ⊢ (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 7438	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑦 +s 𝐶) = (𝐴 +s 𝐶))
13	12	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
14	13	rexsng 4676	. . . . . . . . . . 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 2808	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
17		df-sn 4627	. . . . . . . . 9 ⊢ {(𝐴 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)}
18	16, 17	eqtr4di 2795	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} = {(𝐴 +s 𝐶)})
19		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐴 → (𝐶 +s 𝑦) = (𝐶 +s 𝐴))
20	19	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐴 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
21	20	rexsng 4676	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ No → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
22	2, 21	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
23	7, 2	addscomd 28000	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 +s 𝐴) = (𝐴 +s 𝐶))
24	23	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = (𝐶 +s 𝐴) ↔ 𝑥 = (𝐴 +s 𝐶)))
25	22, 24	bitrd 279	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐴 +s 𝐶)))
26	25	abbidv 2808	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)} = {𝑥 ∣ 𝑥 = (𝐴 +s 𝐶)})
27	26, 17	eqtr4di 2795	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)} = {(𝐴 +s 𝐶)})
28	18, 27	uneq12d 4169	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) = ({(𝐴 +s 𝐶)} ∪ {(𝐴 +s 𝐶)}))
29		unidm 4157	. . . . . . 7 ⊢ ({(𝐴 +s 𝐶)} ∪ {(𝐴 +s 𝐶)}) = {(𝐴 +s 𝐶)}
30	28, 29	eqtrdi 2793	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) = {(𝐴 +s 𝐶)})
31		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐶) = (𝐵 +s 𝐶))
32	31	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
33	32	rexsng 4676	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
34	3, 33	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
35	34	abbidv 2808	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
36		df-sn 4627	. . . . . . . . 9 ⊢ {(𝐵 +s 𝐶)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)}
37	35, 36	eqtr4di 2795	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} = {(𝐵 +s 𝐶)})
38		oveq2 7439	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (𝐶 +s 𝑦) = (𝐶 +s 𝐵))
39	38	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
40	39	rexsng 4676	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
41	3, 40	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
42	7, 3	addscomd 28000	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 +s 𝐵) = (𝐵 +s 𝐶))
43	42	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 = (𝐶 +s 𝐵) ↔ 𝑥 = (𝐵 +s 𝐶)))
44	41, 43	bitrd 279	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐵 +s 𝐶)))
45	44	abbidv 2808	. . . . . . . . 9 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)} = {𝑥 ∣ 𝑥 = (𝐵 +s 𝐶)})
46	45, 36	eqtr4di 2795	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)} = {(𝐵 +s 𝐶)})
47	37, 46	uneq12d 4169	. . . . . . 7 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)}) = ({(𝐵 +s 𝐶)} ∪ {(𝐵 +s 𝐶)}))
48		unidm 4157	. . . . . . 7 ⊢ ({(𝐵 +s 𝐶)} ∪ {(𝐵 +s 𝐶)}) = {(𝐵 +s 𝐶)}
49	47, 48	eqtrdi 2793	. . . . . 6 ⊢ (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)}) = {(𝐵 +s 𝐶)})
50	30, 49	oveq12d 7449	. . . . 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 𝐶)} |s {(𝐵 +s 𝐶)}))
51		2sno 28403	. . . . . . . . 9 ⊢ 2s ∈ No
52	51	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2s ∈ No )
53	52, 2	mulscld 28161	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ∈ No )
54	52, 3	mulscld 28161	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐵) ∈ No )
55		2nns 28402	. . . . . . . . . 10 ⊢ 2s ∈ ℕs
56		nnsgt0 28342	. . . . . . . . . 10 ⊢ (2s ∈ ℕs → 0s <s 2s)
57	55, 56	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → 0s <s 2s)
58	2, 3, 52, 57	sltmul2d 28198	. . . . . . . 8 ⊢ (𝜑 → (𝐴 <s 𝐵 ↔ (2s ·s 𝐴) <s (2s ·s 𝐵)))
59	4, 58	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) <s (2s ·s 𝐵))
60	53, 54, 59	ssltsn 27837	. . . . . 6 ⊢ (𝜑 → {(2s ·s 𝐴)} <<s {(2s ·s 𝐵)})
61	2, 7	addscld 28013	. . . . . . . 8 ⊢ (𝜑 → (𝐴 +s 𝐶) ∈ No )
62		no2times 28401	. . . . . . . . . 10 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
63	2, 62	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
64		slerflex 27808	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → 𝐴 ≤s 𝐴)
65	2, 64	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≤s 𝐴)
66		breq2 5147	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → (𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
67	66	rexsng 4676	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
68	2, 67	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ↔ 𝐴 ≤s 𝐴))
69	65, 68	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥)
70	69	orcd 874	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶))
71		lltropt 27911	. . . . . . . . . . . . 13 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
72	71	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
73		lrcut 27941	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
74	2, 73	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
75	74	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
76		sltrec 27865	. . . . . . . . . . . 12 ⊢ (((( L ‘𝐴) <<s ( R ‘𝐴) ∧ {𝐴} <<s {𝐵}) ∧ (𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)) ∧ 𝐶 = ({𝐴} |s {𝐵}))) → (𝐴 <s 𝐶 ↔ (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶)))
77	72, 5, 75, 10, 76	syl22anc 839	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴 <s 𝐶 ↔ (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶)))
78	70, 77	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 <s 𝐶)
79	2, 7, 2	sltadd2d 28030	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 <s 𝐶 ↔ (𝐴 +s 𝐴) <s (𝐴 +s 𝐶)))
80	78, 79	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐴) <s (𝐴 +s 𝐶))
81	63, 80	eqbrtrd 5165	. . . . . . . 8 ⊢ (𝜑 → (2s ·s 𝐴) <s (𝐴 +s 𝐶))
82	53, 61, 81	sltled 27814	. . . . . . 7 ⊢ (𝜑 → (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
83		ovex 7464	. . . . . . . . . 10 ⊢ (𝐴 +s 𝐶) ∈ V
84		breq2 5147	. . . . . . . . . 10 ⊢ (𝑦 = (𝐴 +s 𝐶) → (𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (𝐴 +s 𝐶)))
85	83, 84	rexsn 4682	. . . . . . . . 9 ⊢ (∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ 𝑥 ≤s (𝐴 +s 𝐶))
86	85	ralbii 3093	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ ∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (𝐴 +s 𝐶))
87		ovex 7464	. . . . . . . . 9 ⊢ (2s ·s 𝐴) ∈ V
88		breq1 5146	. . . . . . . . 9 ⊢ (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s (𝐴 +s 𝐶) ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶)))
89	87, 88	ralsn 4681	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (𝐴 +s 𝐶) ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
90	86, 89	bitri 275	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
91	82, 90	sylibr 234	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦)
92	3, 7	addscld 28013	. . . . . . . 8 ⊢ (𝜑 → (𝐵 +s 𝐶) ∈ No )
93		slerflex 27808	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ No → 𝐵 ≤s 𝐵)
94	3, 93	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ≤s 𝐵)
95		breq1 5146	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → (𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
96	95	rexsng 4676	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ No → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
97	3, 96	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵 ↔ 𝐵 ≤s 𝐵))
98	94, 97	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)
99	98	olcd 875	. . . . . . . . . . 11 ⊢ (𝜑 → (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵))
100		lltropt 27911	. . . . . . . . . . . . 13 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
101	100	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ( L ‘𝐵) <<s ( R ‘𝐵))
102		lrcut 27941	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
103	3, 102	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
104	103	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
105		sltrec 27865	. . . . . . . . . . . 12 ⊢ ((({𝐴} <<s {𝐵} ∧ ( L ‘𝐵) <<s ( R ‘𝐵)) ∧ (𝐶 = ({𝐴} |s {𝐵}) ∧ 𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))) → (𝐶 <s 𝐵 ↔ (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)))
106	5, 101, 10, 104, 105	syl22anc 839	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 <s 𝐵 ↔ (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)))
107	99, 106	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 <s 𝐵)
108	7, 3, 3	sltadd2d 28030	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 <s 𝐵 ↔ (𝐵 +s 𝐶) <s (𝐵 +s 𝐵)))
109	107, 108	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 +s 𝐶) <s (𝐵 +s 𝐵))
110		no2times 28401	. . . . . . . . . 10 ⊢ (𝐵 ∈ No → (2s ·s 𝐵) = (𝐵 +s 𝐵))
111	3, 110	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (2s ·s 𝐵) = (𝐵 +s 𝐵))
112	109, 111	breqtrrd 5171	. . . . . . . 8 ⊢ (𝜑 → (𝐵 +s 𝐶) <s (2s ·s 𝐵))
113	92, 54, 112	sltled 27814	. . . . . . 7 ⊢ (𝜑 → (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
114		ovex 7464	. . . . . . . . . 10 ⊢ (𝐵 +s 𝐶) ∈ V
115		breq1 5146	. . . . . . . . . 10 ⊢ (𝑦 = (𝐵 +s 𝐶) → (𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s 𝑥))
116	114, 115	rexsn 4682	. . . . . . . . 9 ⊢ (∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s 𝑥)
117	116	ralbii 3093	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ ∀𝑥 ∈ {(2s ·s 𝐵)} (𝐵 +s 𝐶) ≤s 𝑥)
118		ovex 7464	. . . . . . . . 9 ⊢ (2s ·s 𝐵) ∈ V
119		breq2 5147	. . . . . . . . 9 ⊢ (𝑥 = (2s ·s 𝐵) → ((𝐵 +s 𝐶) ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵)))
120	118, 119	ralsn 4681	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(2s ·s 𝐵)} (𝐵 +s 𝐶) ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
121	117, 120	bitri 275	. . . . . . 7 ⊢ (∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
122	113, 121	sylibr 234	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥)
123	2, 3	addscld 28013	. . . . . . . 8 ⊢ (𝜑 → (𝐴 +s 𝐵) ∈ No )
124	7, 3, 2	sltadd2d 28030	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐴 +s 𝐵)))
125	107, 124	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝐴 +s 𝐶) <s (𝐴 +s 𝐵))
126	61, 123, 125	ssltsn 27837	. . . . . . 7 ⊢ (𝜑 → {(𝐴 +s 𝐶)} <<s {(𝐴 +s 𝐵)})
127		halfcut.4	. . . . . . . 8 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))
128	127	sneqd 4638	. . . . . . 7 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} = {(𝐴 +s 𝐵)})
129	126, 128	breqtrrd 5171	. . . . . 6 ⊢ (𝜑 → {(𝐴 +s 𝐶)} <<s {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})})
130	2, 3	addscomd 28000	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
131	2, 7, 3	sltadd2d 28030	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 <s 𝐶 ↔ (𝐵 +s 𝐴) <s (𝐵 +s 𝐶)))
132	78, 131	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 +s 𝐴) <s (𝐵 +s 𝐶))
133	130, 132	eqbrtrd 5165	. . . . . . . 8 ⊢ (𝜑 → (𝐴 +s 𝐵) <s (𝐵 +s 𝐶))
134	123, 92, 133	ssltsn 27837	. . . . . . 7 ⊢ (𝜑 → {(𝐴 +s 𝐵)} <<s {(𝐵 +s 𝐶)})
135	128, 134	eqbrtrd 5165	. . . . . 6 ⊢ (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} <<s {(𝐵 +s 𝐶)})
136	60, 91, 122, 129, 135	cofcut1d 27955	. . . . 5 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = ({(𝐴 +s 𝐶)} |s {(𝐵 +s 𝐶)}))
137	50, 136, 127	3eqtr2d 2783	. . . 4 ⊢ (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))
138	9, 11, 137	3eqtrd 2781	. . 3 ⊢ (𝜑 → (2s ·s 𝐶) = (𝐴 +s 𝐵))
139		2ne0s 28404	. . . . 5 ⊢ 2s ≠ 0s
140	139	a1i 11	. . . 4 ⊢ (𝜑 → 2s ≠ 0s )
141	123, 7, 52, 140	divsmuld 28246	. . 3 ⊢ (𝜑 → (((𝐴 +s 𝐵) /su 2s) = 𝐶 ↔ (2s ·s 𝐶) = (𝐴 +s 𝐵)))
142	138, 141	mpbird 257	. 2 ⊢ (𝜑 → ((𝐴 +s 𝐵) /su 2s) = 𝐶)
143	142	eqcomd 2743	1 ⊢ (𝜑 → 𝐶 = ((𝐴 +s 𝐵) /su 2s))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1540   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ∪ cun 3949  {csn 4626   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   ≤s csle 27789   <<s csslt 27825   |s cscut 27827   0_s c0s 27867   L cleft 27884   R cright 27885   +_s cadds 27992   ·_s cmuls 28132   /_su cdivs 28213  ℕ_scnns 28319  2_sc2s 28394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-dc 10486

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214  df-n0s 28320  df-nns 28321  df-2s 28395

This theorem is referenced by:  cutpw2  28417  addhalfcut  28419  pw2cut  28420