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Theorem halfcut 28431
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 (𝜑𝐴 No )
halfcut.2 (𝜑𝐵 No )
halfcut.3 (𝜑𝐴 <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 (𝜑𝐶 = ((𝐴 +s 𝐵) /su 2s))

Proof of Theorem halfcut
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 halfcut.5 . . . . . 6 𝐶 = ({𝐴} |s {𝐵})
2 halfcut.1 . . . . . . . 8 (𝜑𝐴 No )
3 halfcut.2 . . . . . . . 8 (𝜑𝐵 No )
4 halfcut.3 . . . . . . . 8 (𝜑𝐴 <s 𝐵)
52, 3, 4ssltsn 27852 . . . . . . 7 (𝜑 → {𝐴} <<s {𝐵})
65scutcld 27863 . . . . . 6 (𝜑 → ({𝐴} |s {𝐵}) ∈ No )
71, 6eqeltrid 2843 . . . . 5 (𝜑𝐶 No )
8 no2times 28416 . . . . 5 (𝐶 No → (2s ·s 𝐶) = (𝐶 +s 𝐶))
97, 8syl 17 . . . 4 (𝜑 → (2s ·s 𝐶) = (𝐶 +s 𝐶))
101a1i 11 . . . . 5 (𝜑𝐶 = ({𝐴} |s {𝐵}))
115, 5, 10, 10addsunif 28050 . . . 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))
1312eqeq2d 2746 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
1413rexsng 4681 . . . . . . . . . . 11 (𝐴 No → (∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
152, 14syl 17 . . . . . . . . . 10 (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐴 +s 𝐶)))
1615abbidv 2806 . . . . . . . . 9 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} = {𝑥𝑥 = (𝐴 +s 𝐶)})
17 df-sn 4632 . . . . . . . . 9 {(𝐴 +s 𝐶)} = {𝑥𝑥 = (𝐴 +s 𝐶)}
1816, 17eqtr4di 2793 . . . . . . . 8 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} = {(𝐴 +s 𝐶)})
19 oveq2 7439 . . . . . . . . . . . . . 14 (𝑦 = 𝐴 → (𝐶 +s 𝑦) = (𝐶 +s 𝐴))
2019eqeq2d 2746 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
2120rexsng 4681 . . . . . . . . . . . 12 (𝐴 No → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
222, 21syl 17 . . . . . . . . . . 11 (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐴)))
237, 2addscomd 28015 . . . . . . . . . . . 12 (𝜑 → (𝐶 +s 𝐴) = (𝐴 +s 𝐶))
2423eqeq2d 2746 . . . . . . . . . . 11 (𝜑 → (𝑥 = (𝐶 +s 𝐴) ↔ 𝑥 = (𝐴 +s 𝐶)))
2522, 24bitrd 279 . . . . . . . . . 10 (𝜑 → (∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐴 +s 𝐶)))
2625abbidv 2806 . . . . . . . . 9 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)} = {𝑥𝑥 = (𝐴 +s 𝐶)})
2726, 17eqtr4di 2793 . . . . . . . 8 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)} = {(𝐴 +s 𝐶)})
2818, 27uneq12d 4179 . . . . . . 7 (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐴}𝑥 = (𝐶 +s 𝑦)}) = ({(𝐴 +s 𝐶)} ∪ {(𝐴 +s 𝐶)}))
29 unidm 4167 . . . . . . 7 ({(𝐴 +s 𝐶)} ∪ {(𝐴 +s 𝐶)}) = {(𝐴 +s 𝐶)}
3028, 29eqtrdi 2791 . . . . . 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))
3231eqeq2d 2746 . . . . . . . . . . . 12 (𝑦 = 𝐵 → (𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
3332rexsng 4681 . . . . . . . . . . 11 (𝐵 No → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
343, 33syl 17 . . . . . . . . . 10 (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶) ↔ 𝑥 = (𝐵 +s 𝐶)))
3534abbidv 2806 . . . . . . . . 9 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} = {𝑥𝑥 = (𝐵 +s 𝐶)})
36 df-sn 4632 . . . . . . . . 9 {(𝐵 +s 𝐶)} = {𝑥𝑥 = (𝐵 +s 𝐶)}
3735, 36eqtr4di 2793 . . . . . . . 8 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} = {(𝐵 +s 𝐶)})
38 oveq2 7439 . . . . . . . . . . . . . 14 (𝑦 = 𝐵 → (𝐶 +s 𝑦) = (𝐶 +s 𝐵))
3938eqeq2d 2746 . . . . . . . . . . . . 13 (𝑦 = 𝐵 → (𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
4039rexsng 4681 . . . . . . . . . . . 12 (𝐵 No → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
413, 40syl 17 . . . . . . . . . . 11 (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐶 +s 𝐵)))
427, 3addscomd 28015 . . . . . . . . . . . 12 (𝜑 → (𝐶 +s 𝐵) = (𝐵 +s 𝐶))
4342eqeq2d 2746 . . . . . . . . . . 11 (𝜑 → (𝑥 = (𝐶 +s 𝐵) ↔ 𝑥 = (𝐵 +s 𝐶)))
4441, 43bitrd 279 . . . . . . . . . 10 (𝜑 → (∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦) ↔ 𝑥 = (𝐵 +s 𝐶)))
4544abbidv 2806 . . . . . . . . 9 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)} = {𝑥𝑥 = (𝐵 +s 𝐶)})
4645, 36eqtr4di 2793 . . . . . . . 8 (𝜑 → {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)} = {(𝐵 +s 𝐶)})
4737, 46uneq12d 4179 . . . . . . 7 (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)}) = ({(𝐵 +s 𝐶)} ∪ {(𝐵 +s 𝐶)}))
48 unidm 4167 . . . . . . 7 ({(𝐵 +s 𝐶)} ∪ {(𝐵 +s 𝐶)}) = {(𝐵 +s 𝐶)}
4947, 48eqtrdi 2791 . . . . . 6 (𝜑 → ({𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝑦 +s 𝐶)} ∪ {𝑥 ∣ ∃𝑦 ∈ {𝐵}𝑥 = (𝐶 +s 𝑦)}) = {(𝐵 +s 𝐶)})
5030, 49oveq12d 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 28418 . . . . . . . . 9 2s No
5251a1i 11 . . . . . . . 8 (𝜑 → 2s No )
5352, 2mulscld 28176 . . . . . . 7 (𝜑 → (2s ·s 𝐴) ∈ No )
5452, 3mulscld 28176 . . . . . . 7 (𝜑 → (2s ·s 𝐵) ∈ No )
55 2nns 28417 . . . . . . . . . 10 2s ∈ ℕs
56 nnsgt0 28357 . . . . . . . . . 10 (2s ∈ ℕs → 0s <s 2s)
5755, 56mp1i 13 . . . . . . . . 9 (𝜑 → 0s <s 2s)
582, 3, 52, 57sltmul2d 28213 . . . . . . . 8 (𝜑 → (𝐴 <s 𝐵 ↔ (2s ·s 𝐴) <s (2s ·s 𝐵)))
594, 58mpbid 232 . . . . . . 7 (𝜑 → (2s ·s 𝐴) <s (2s ·s 𝐵))
6053, 54, 59ssltsn 27852 . . . . . 6 (𝜑 → {(2s ·s 𝐴)} <<s {(2s ·s 𝐵)})
612, 7addscld 28028 . . . . . . . 8 (𝜑 → (𝐴 +s 𝐶) ∈ No )
62 no2times 28416 . . . . . . . . . 10 (𝐴 No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
632, 62syl 17 . . . . . . . . 9 (𝜑 → (2s ·s 𝐴) = (𝐴 +s 𝐴))
64 slerflex 27823 . . . . . . . . . . . . . 14 (𝐴 No 𝐴 ≤s 𝐴)
652, 64syl 17 . . . . . . . . . . . . 13 (𝜑𝐴 ≤s 𝐴)
66 breq2 5152 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 → (𝐴 ≤s 𝑥𝐴 ≤s 𝐴))
6766rexsng 4681 . . . . . . . . . . . . . 14 (𝐴 No → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥𝐴 ≤s 𝐴))
682, 67syl 17 . . . . . . . . . . . . 13 (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥𝐴 ≤s 𝐴))
6965, 68mpbird 257 . . . . . . . . . . . 12 (𝜑 → ∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥)
7069orcd 873 . . . . . . . . . . 11 (𝜑 → (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶))
71 lltropt 27926 . . . . . . . . . . . . 13 ( L ‘𝐴) <<s ( R ‘𝐴)
7271a1i 11 . . . . . . . . . . . 12 (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
73 lrcut 27956 . . . . . . . . . . . . . 14 (𝐴 No → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
742, 73syl 17 . . . . . . . . . . . . 13 (𝜑 → (( L ‘𝐴) |s ( R ‘𝐴)) = 𝐴)
7574eqcomd 2741 . . . . . . . . . . . 12 (𝜑𝐴 = (( L ‘𝐴) |s ( R ‘𝐴)))
76 sltrec 27880 . . . . . . . . . . . 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)))
7772, 5, 75, 10, 76syl22anc 839 . . . . . . . . . . 11 (𝜑 → (𝐴 <s 𝐶 ↔ (∃𝑥 ∈ {𝐴}𝐴 ≤s 𝑥 ∨ ∃𝑦 ∈ ( R ‘𝐴)𝑦 ≤s 𝐶)))
7870, 77mpbird 257 . . . . . . . . . 10 (𝜑𝐴 <s 𝐶)
792, 7, 2sltadd2d 28045 . . . . . . . . . 10 (𝜑 → (𝐴 <s 𝐶 ↔ (𝐴 +s 𝐴) <s (𝐴 +s 𝐶)))
8078, 79mpbid 232 . . . . . . . . 9 (𝜑 → (𝐴 +s 𝐴) <s (𝐴 +s 𝐶))
8163, 80eqbrtrd 5170 . . . . . . . 8 (𝜑 → (2s ·s 𝐴) <s (𝐴 +s 𝐶))
8253, 61, 81sltled 27829 . . . . . . 7 (𝜑 → (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
83 ovex 7464 . . . . . . . . . 10 (𝐴 +s 𝐶) ∈ V
84 breq2 5152 . . . . . . . . . 10 (𝑦 = (𝐴 +s 𝐶) → (𝑥 ≤s 𝑦𝑥 ≤s (𝐴 +s 𝐶)))
8583, 84rexsn 4687 . . . . . . . . 9 (∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦𝑥 ≤s (𝐴 +s 𝐶))
8685ralbii 3091 . . . . . . . 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 5151 . . . . . . . . 9 (𝑥 = (2s ·s 𝐴) → (𝑥 ≤s (𝐴 +s 𝐶) ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶)))
8987, 88ralsn 4686 . . . . . . . 8 (∀𝑥 ∈ {(2s ·s 𝐴)}𝑥 ≤s (𝐴 +s 𝐶) ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
9086, 89bitri 275 . . . . . . 7 (∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦 ↔ (2s ·s 𝐴) ≤s (𝐴 +s 𝐶))
9182, 90sylibr 234 . . . . . 6 (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐴)}∃𝑦 ∈ {(𝐴 +s 𝐶)}𝑥 ≤s 𝑦)
923, 7addscld 28028 . . . . . . . 8 (𝜑 → (𝐵 +s 𝐶) ∈ No )
93 slerflex 27823 . . . . . . . . . . . . . 14 (𝐵 No 𝐵 ≤s 𝐵)
943, 93syl 17 . . . . . . . . . . . . 13 (𝜑𝐵 ≤s 𝐵)
95 breq1 5151 . . . . . . . . . . . . . . 15 (𝑦 = 𝐵 → (𝑦 ≤s 𝐵𝐵 ≤s 𝐵))
9695rexsng 4681 . . . . . . . . . . . . . 14 (𝐵 No → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵𝐵 ≤s 𝐵))
973, 96syl 17 . . . . . . . . . . . . 13 (𝜑 → (∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵𝐵 ≤s 𝐵))
9894, 97mpbird 257 . . . . . . . . . . . 12 (𝜑 → ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)
9998olcd 874 . . . . . . . . . . 11 (𝜑 → (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵))
100 lltropt 27926 . . . . . . . . . . . . 13 ( L ‘𝐵) <<s ( R ‘𝐵)
101100a1i 11 . . . . . . . . . . . 12 (𝜑 → ( L ‘𝐵) <<s ( R ‘𝐵))
102 lrcut 27956 . . . . . . . . . . . . . 14 (𝐵 No → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
1033, 102syl 17 . . . . . . . . . . . . 13 (𝜑 → (( L ‘𝐵) |s ( R ‘𝐵)) = 𝐵)
104103eqcomd 2741 . . . . . . . . . . . 12 (𝜑𝐵 = (( L ‘𝐵) |s ( R ‘𝐵)))
105 sltrec 27880 . . . . . . . . . . . 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)))
1065, 101, 10, 104, 105syl22anc 839 . . . . . . . . . . 11 (𝜑 → (𝐶 <s 𝐵 ↔ (∃𝑥 ∈ ( L ‘𝐵)𝐶 ≤s 𝑥 ∨ ∃𝑦 ∈ {𝐵}𝑦 ≤s 𝐵)))
10799, 106mpbird 257 . . . . . . . . . 10 (𝜑𝐶 <s 𝐵)
1087, 3, 3sltadd2d 28045 . . . . . . . . . 10 (𝜑 → (𝐶 <s 𝐵 ↔ (𝐵 +s 𝐶) <s (𝐵 +s 𝐵)))
109107, 108mpbid 232 . . . . . . . . 9 (𝜑 → (𝐵 +s 𝐶) <s (𝐵 +s 𝐵))
110 no2times 28416 . . . . . . . . . 10 (𝐵 No → (2s ·s 𝐵) = (𝐵 +s 𝐵))
1113, 110syl 17 . . . . . . . . 9 (𝜑 → (2s ·s 𝐵) = (𝐵 +s 𝐵))
112109, 111breqtrrd 5176 . . . . . . . 8 (𝜑 → (𝐵 +s 𝐶) <s (2s ·s 𝐵))
11392, 54, 112sltled 27829 . . . . . . 7 (𝜑 → (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
114 ovex 7464 . . . . . . . . . 10 (𝐵 +s 𝐶) ∈ V
115 breq1 5151 . . . . . . . . . 10 (𝑦 = (𝐵 +s 𝐶) → (𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s 𝑥))
116114, 115rexsn 4687 . . . . . . . . 9 (∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s 𝑥)
117116ralbii 3091 . . . . . . . 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 5152 . . . . . . . . 9 (𝑥 = (2s ·s 𝐵) → ((𝐵 +s 𝐶) ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵)))
120118, 119ralsn 4686 . . . . . . . 8 (∀𝑥 ∈ {(2s ·s 𝐵)} (𝐵 +s 𝐶) ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
121117, 120bitri 275 . . . . . . 7 (∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥 ↔ (𝐵 +s 𝐶) ≤s (2s ·s 𝐵))
122113, 121sylibr 234 . . . . . 6 (𝜑 → ∀𝑥 ∈ {(2s ·s 𝐵)}∃𝑦 ∈ {(𝐵 +s 𝐶)}𝑦 ≤s 𝑥)
1232, 3addscld 28028 . . . . . . . 8 (𝜑 → (𝐴 +s 𝐵) ∈ No )
1247, 3, 2sltadd2d 28045 . . . . . . . . 9 (𝜑 → (𝐶 <s 𝐵 ↔ (𝐴 +s 𝐶) <s (𝐴 +s 𝐵)))
125107, 124mpbid 232 . . . . . . . 8 (𝜑 → (𝐴 +s 𝐶) <s (𝐴 +s 𝐵))
12661, 123, 125ssltsn 27852 . . . . . . 7 (𝜑 → {(𝐴 +s 𝐶)} <<s {(𝐴 +s 𝐵)})
127 halfcut.4 . . . . . . . 8 (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))
128127sneqd 4643 . . . . . . 7 (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} = {(𝐴 +s 𝐵)})
129126, 128breqtrrd 5176 . . . . . 6 (𝜑 → {(𝐴 +s 𝐶)} <<s {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})})
1302, 3addscomd 28015 . . . . . . . . 9 (𝜑 → (𝐴 +s 𝐵) = (𝐵 +s 𝐴))
1312, 7, 3sltadd2d 28045 . . . . . . . . . 10 (𝜑 → (𝐴 <s 𝐶 ↔ (𝐵 +s 𝐴) <s (𝐵 +s 𝐶)))
13278, 131mpbid 232 . . . . . . . . 9 (𝜑 → (𝐵 +s 𝐴) <s (𝐵 +s 𝐶))
133130, 132eqbrtrd 5170 . . . . . . . 8 (𝜑 → (𝐴 +s 𝐵) <s (𝐵 +s 𝐶))
134123, 92, 133ssltsn 27852 . . . . . . 7 (𝜑 → {(𝐴 +s 𝐵)} <<s {(𝐵 +s 𝐶)})
135128, 134eqbrtrd 5170 . . . . . 6 (𝜑 → {({(2s ·s 𝐴)} |s {(2s ·s 𝐵)})} <<s {(𝐵 +s 𝐶)})
13660, 91, 122, 129, 135cofcut1d 27970 . . . . 5 (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = ({(𝐴 +s 𝐶)} |s {(𝐵 +s 𝐶)}))
13750, 136, 1273eqtr2d 2781 . . . 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))
1389, 11, 1373eqtrd 2779 . . 3 (𝜑 → (2s ·s 𝐶) = (𝐴 +s 𝐵))
139 2ne0s 28419 . . . . 5 2s ≠ 0s
140139a1i 11 . . . 4 (𝜑 → 2s ≠ 0s )
141123, 7, 52, 140divsmuld 28261 . . 3 (𝜑 → (((𝐴 +s 𝐵) /su 2s) = 𝐶 ↔ (2s ·s 𝐶) = (𝐴 +s 𝐵)))
142138, 141mpbird 257 . 2 (𝜑 → ((𝐴 +s 𝐵) /su 2s) = 𝐶)
143142eqcomd 2741 1 (𝜑𝐶 = ((𝐴 +s 𝐵) /su 2s))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  cun 3961  {csn 4631   class class class wbr 5148  cfv 6563  (class class class)co 7431   No csur 27699   <s cslt 27700   ≤s csle 27804   <<s csslt 27840   |s cscut 27842   0s c0s 27882   L cleft 27899   R cright 27900   +s cadds 28007   ·s cmuls 28147   /su cdivs 28228  scnns 28334  2sc2s 28409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229  df-n0s 28335  df-nns 28336  df-2s 28410
This theorem is referenced by:  cutpw2  28432  addhalfcut  28434  pw2cut  28435
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