Theorem mulsrid 28121

Description: Surreal one is a right identity element for multiplication. (Contributed by Scott Fenton, 4-Feb-2025.)

Assertion

Ref	Expression
mulsrid	⊢ (𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)

Proof of Theorem mulsrid

Dummy variables 𝑥 𝑥_𝑂 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7375	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 1s ) = (𝑥𝑂 ·s 1s ))
2		id 22	. . 3 ⊢ (𝑥 = 𝑥𝑂 → 𝑥 = 𝑥𝑂)
3	1, 2	eqeq12d 2753	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝑥𝑂 ·s 1s ) = 𝑥𝑂))
4		oveq1 7375	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 1s ) = (𝐴 ·s 1s ))
5		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6	4, 5	eqeq12d 2753	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝐴 ·s 1s ) = 𝐴))
7		1no 27818	. . . . . 6 ⊢ 1s ∈ No
8		mulsval 28117	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → (𝑥 ·s 1s ) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
9	7, 8	mpan2 692	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 ·s 1s ) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
10	9	adantr 480	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑥 ·s 1s ) = (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))
11		elun1 4136	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ ( L ‘𝑥) → 𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
12		oveq1 7375	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑝 → (𝑥𝑂 ·s 1s ) = (𝑝 ·s 1s ))
13		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑝 → 𝑥𝑂 = 𝑝)
14	12, 13	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑝 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑝 ·s 1s ) = 𝑝))
15	14	rspcva 3576	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑝 ·s 1s ) = 𝑝)
16	11, 15	sylan 581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ∈ ( L ‘𝑥) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑝 ·s 1s ) = 𝑝)
17	16	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂 ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 1s ) = 𝑝)
18	17	adantll 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 1s ) = 𝑝)
19		muls01 28120	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ No → (𝑥 ·s 0s ) = 0s )
20	19	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑥 ·s 0s ) = 0s )
21	20	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑥 ·s 0s ) = 0s )
22	18, 21	oveq12d 7386	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → ((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) = (𝑝 +s 0s ))
23		leftno 27885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ( L ‘𝑥) → 𝑝 ∈ No )
24	23	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → 𝑝 ∈ No )
25	24	addsridd 27973	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 +s 0s ) = 𝑝)
26	22, 25	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → ((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) = 𝑝)
27		muls01 28120	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ No → (𝑝 ·s 0s ) = 0s )
28	24, 27	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 0s ) = 0s )
29	26, 28	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )) = (𝑝 -s 0s ))
30		subsid1 28076	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ No → (𝑝 -s 0s ) = 𝑝)
31	24, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 -s 0s ) = 𝑝)
32	29, 31	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )) = 𝑝)
33	32	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )) ↔ 𝑎 = 𝑝))
34		equcom 2020	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑝 ↔ 𝑝 = 𝑎)
35	33, 34	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )) ↔ 𝑝 = 𝑎))
36	35	rexbidva 3160	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )) ↔ ∃𝑝 ∈ ( L ‘𝑥)𝑝 = 𝑎))
37		left1s 27903	. . . . . . . . . . . 12 ⊢ ( L ‘ 1s ) = { 0s }
38	37	rexeqi 3297	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑞 ∈ { 0s }𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
39		0no 27817	. . . . . . . . . . . . 13 ⊢ 0s ∈ No
40	39	elexi 3465	. . . . . . . . . . . 12 ⊢ 0s ∈ V
41		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 0s → (𝑥 ·s 𝑞) = (𝑥 ·s 0s ))
42	41	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 0s → ((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) = ((𝑝 ·s 1s ) +s (𝑥 ·s 0s )))
43		oveq2 7376	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 0s → (𝑝 ·s 𝑞) = (𝑝 ·s 0s ))
44	42, 43	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (𝑞 = 0s → (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )))
45	44	eqeq2d 2748	. . . . . . . . . . . 12 ⊢ (𝑞 = 0s → (𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s ))))
46	40, 45	rexsn 4641	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ { 0s }𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )))
47	38, 46	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )))
48	47	rexbii 3085	. . . . . . . . 9 ⊢ (∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑝 ∈ ( L ‘𝑥)𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑝 ·s 0s )))
49		risset 3213	. . . . . . . . 9 ⊢ (𝑎 ∈ ( L ‘𝑥) ↔ ∃𝑝 ∈ ( L ‘𝑥)𝑝 = 𝑎)
50	36, 48, 49	3bitr4g 314	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑎 ∈ ( L ‘𝑥)))
51	50	eqabcdv 2871	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = ( L ‘𝑥))
52		rex0 4314	. . . . . . . . . . . 12 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
53		right1s 27904	. . . . . . . . . . . . 13 ⊢ ( R ‘ 1s ) = ∅
54	53	rexeqi 3297	. . . . . . . . . . . 12 ⊢ (∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
55	52, 54	mtbir 323	. . . . . . . . . . 11 ⊢ ¬ ∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
56	55	a1i 11	. . . . . . . . . 10 ⊢ (𝑟 ∈ ( R ‘𝑥) → ¬ ∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
57	56	nrex 3066	. . . . . . . . 9 ⊢ ¬ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
58	57	abf 4360	. . . . . . . 8 ⊢ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = ∅
59	58	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = ∅)
60	51, 59	uneq12d 4123	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) = (( L ‘𝑥) ∪ ∅))
61		un0 4348	. . . . . 6 ⊢ (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
62	60, 61	eqtrdi 2788	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → ({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) = ( L ‘𝑥))
63		rex0 4314	. . . . . . . . . . . 12 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
64	53	rexeqi 3297	. . . . . . . . . . . 12 ⊢ (∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
65	63, 64	mtbir 323	. . . . . . . . . . 11 ⊢ ¬ ∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
66	65	a1i 11	. . . . . . . . . 10 ⊢ (𝑡 ∈ ( L ‘𝑥) → ¬ ∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
67	66	nrex 3066	. . . . . . . . 9 ⊢ ¬ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
68	67	abf 4360	. . . . . . . 8 ⊢ {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = ∅
69	68	a1i 11	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = ∅)
70		elun2 4137	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ( R ‘𝑥) → 𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
71		oveq1 7375	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑣 → (𝑥𝑂 ·s 1s ) = (𝑣 ·s 1s ))
72		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑣 → 𝑥𝑂 = 𝑣)
73	71, 72	eqeq12d 2753	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑣 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑣 ·s 1s ) = 𝑣))
74	73	rspcva 3576	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑣 ·s 1s ) = 𝑣)
75	70, 74	sylan 581	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ ( R ‘𝑥) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑣 ·s 1s ) = 𝑣)
76	75	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂 ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 1s ) = 𝑣)
77	76	adantll 715	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 1s ) = 𝑣)
78	20	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑥 ·s 0s ) = 0s )
79	77, 78	oveq12d 7386	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → ((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) = (𝑣 +s 0s ))
80		rightno 27886	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ( R ‘𝑥) → 𝑣 ∈ No )
81	80	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → 𝑣 ∈ No )
82	81	addsridd 27973	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 +s 0s ) = 𝑣)
83	79, 82	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → ((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) = 𝑣)
84		muls01 28120	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ No → (𝑣 ·s 0s ) = 0s )
85	81, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 0s ) = 0s )
86	83, 85	oveq12d 7386	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑣 ·s 0s )) = (𝑣 -s 0s ))
87		subsid1 28076	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ No → (𝑣 -s 0s ) = 𝑣)
88	81, 87	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 -s 0s ) = 𝑣)
89	86, 88	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑣 ·s 0s )) = 𝑣)
90	89	eqeq2d 2748	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑣 ·s 0s )) ↔ 𝑑 = 𝑣))
91	37	rexeqi 3297	. . . . . . . . . . . 12 ⊢ (∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑤 ∈ { 0s }𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
92		oveq2 7376	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 0s → (𝑥 ·s 𝑤) = (𝑥 ·s 0s ))
93	92	oveq2d 7384	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 0s → ((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) = ((𝑣 ·s 1s ) +s (𝑥 ·s 0s )))
94		oveq2 7376	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 0s → (𝑣 ·s 𝑤) = (𝑣 ·s 0s ))
95	93, 94	oveq12d 7386	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 0s → (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) = (((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑣 ·s 0s )))
96	95	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑤 = 0s → (𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑣 ·s 0s ))))
97	40, 96	rexsn 4641	. . . . . . . . . . . 12 ⊢ (∃𝑤 ∈ { 0s }𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑣 ·s 0s )))
98	91, 97	bitri 275	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) -s (𝑣 ·s 0s )))
99		equcom 2020	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑑 ↔ 𝑑 = 𝑣)
100	90, 98, 99	3bitr4g 314	. . . . . . . . . 10 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑣 = 𝑑))
101	100	rexbidva 3160	. . . . . . . . 9 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑣 ∈ ( R ‘𝑥)𝑣 = 𝑑))
102		risset 3213	. . . . . . . . 9 ⊢ (𝑑 ∈ ( R ‘𝑥) ↔ ∃𝑣 ∈ ( R ‘𝑥)𝑣 = 𝑑)
103	101, 102	bitr4di 289	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑑 ∈ ( R ‘𝑥)))
104	103	eqabcdv 2871	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))} = ( R ‘𝑥))
105	69, 104	uneq12d 4123	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) = (∅ ∪ ( R ‘𝑥)))
106		0un 4350	. . . . . 6 ⊢ (∅ ∪ ( R ‘𝑥)) = ( R ‘𝑥)
107	105, 106	eqtrdi 2788	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) = ( R ‘𝑥))
108	62, 107	oveq12d 7386	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (({𝑎 ∣ ∃𝑝 ∈ ( L ‘𝑥)∃𝑞 ∈ ( L ‘ 1s )𝑎 = (((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ ( R ‘𝑥)∃𝑤 ∈ ( L ‘ 1s )𝑑 = (((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) = (( L ‘𝑥) |s ( R ‘𝑥)))
109		lrcut 27912	. . . . 5 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
110	109	adantr 480	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
111	10, 108, 110	3eqtrd 2776	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑥 ·s 1s ) = 𝑥)
112	111	ex 412	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂 → (𝑥 ·s 1s ) = 𝑥))
113	3, 6, 112	noinds 27953	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∪ cun 3901  ∅c0 4287  {csn 4582  ‘cfv 6500  (class class class)co 7368   No csur 27619   |s ccuts 27767   0_s c0s 27813   1_s c1s 27814   L cleft 27833   R cright 27834   +_s cadds 27967   -_s csubs 28028   ·_s cmuls 28114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115

This theorem is referenced by:  mulsridd  28122  mulslid  28150  n0seo  28429  zseo  28430  pw2recs  28446  addhalfcut  28467  pw2cutp1  28469  bdaypw2n0bndlem  28471  bdayfinbndlem1  28475  remulscllem1  28508