Theorem mulsrid 28109

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 7365	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 1s ) = (𝑥𝑂 ·s 1s ))
2		id 22	. . 3 ⊢ (𝑥 = 𝑥𝑂 → 𝑥 = 𝑥𝑂)
3	1, 2	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝑥𝑂 ·s 1s ) = 𝑥𝑂))
4		oveq1 7365	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 1s ) = (𝐴 ·s 1s ))
5		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6	4, 5	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝐴 ·s 1s ) = 𝐴))
7		1no 27806	. . . . . 6 ⊢ 1s ∈ No
8		mulsval 28105	. . . . . 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 691	. . . . 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 4134	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ ( L ‘𝑥) → 𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
12		oveq1 7365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑝 → (𝑥𝑂 ·s 1s ) = (𝑝 ·s 1s ))
13		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑝 → 𝑥𝑂 = 𝑝)
14	12, 13	eqeq12d 2752	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑝 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑝 ·s 1s ) = 𝑝))
15	14	rspcva 3574	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑝 ·s 1s ) = 𝑝)
16	11, 15	sylan 580	. . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 1s ) = 𝑝)
19		muls01 28108	. . . . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 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 27873	. . . . . . . . . . . . . . . . 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 27961	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 +s 0s ) = 𝑝)
26	22, 25	eqtrd 2771	. . . . . . . . . . . . . 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 28108	. . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . 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 28064	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 2747	. . . . . . . . . . 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 2019	. . . . . . . . . . 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 3158	. . . . . . . . 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 27891	. . . . . . . . . . . 12 ⊢ ( L ‘ 1s ) = { 0s }
38	37	rexeqi 3295	. . . . . . . . . . 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 27805	. . . . . . . . . . . . 13 ⊢ 0s ∈ No
40	39	elexi 3463	. . . . . . . . . . . 12 ⊢ 0s ∈ V
41		oveq2 7366	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 0s → (𝑥 ·s 𝑞) = (𝑥 ·s 0s ))
42	41	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 0s → ((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) = ((𝑝 ·s 1s ) +s (𝑥 ·s 0s )))
43		oveq2 7366	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 0s → (𝑝 ·s 𝑞) = (𝑝 ·s 0s ))
44	42, 43	oveq12d 7376	. . . . . . . . . . . . 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 2747	. . . . . . . . . . . 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 4639	. . . . . . . . . . 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 3083	. . . . . . . . 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 3211	. . . . . . . . 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 2870	. . . . . . 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 4312	. . . . . . . . . . . 12 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
53		right1s 27892	. . . . . . . . . . . . 13 ⊢ ( R ‘ 1s ) = ∅
54	53	rexeqi 3295	. . . . . . . . . . . 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 3064	. . . . . . . . 9 ⊢ ¬ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
58	57	abf 4358	. . . . . . . 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 4121	. . . . . 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 4346	. . . . . 6 ⊢ (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
62	60, 61	eqtrdi 2787	. . . . 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 4312	. . . . . . . . . . . 12 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
64	53	rexeqi 3295	. . . . . . . . . . . 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 3064	. . . . . . . . 9 ⊢ ¬ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
68	67	abf 4358	. . . . . . . 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 4135	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ( R ‘𝑥) → 𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
71		oveq1 7365	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑣 → (𝑥𝑂 ·s 1s ) = (𝑣 ·s 1s ))
72		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑣 → 𝑥𝑂 = 𝑣)
73	71, 72	eqeq12d 2752	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑣 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑣 ·s 1s ) = 𝑣))
74	73	rspcva 3574	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑣 ·s 1s ) = 𝑣)
75	70, 74	sylan 580	. . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . . . 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 27874	. . . . . . . . . . . . . . . . 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 27961	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 +s 0s ) = 𝑣)
83	79, 82	eqtrd 2771	. . . . . . . . . . . . . 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 28108	. . . . . . . . . . . . . . 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 7376	. . . . . . . . . . . . 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 28064	. . . . . . . . . . . . . 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 2771	. . . . . . . . . . . 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 2747	. . . . . . . . . . 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 3295	. . . . . . . . . . . 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 7366	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 0s → (𝑥 ·s 𝑤) = (𝑥 ·s 0s ))
93	92	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 0s → ((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) = ((𝑣 ·s 1s ) +s (𝑥 ·s 0s )))
94		oveq2 7366	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 0s → (𝑣 ·s 𝑤) = (𝑣 ·s 0s ))
95	93, 94	oveq12d 7376	. . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . 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 4639	. . . . . . . . . . . 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 2019	. . . . . . . . . . 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 3158	. . . . . . . . 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 3211	. . . . . . . . 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 2870	. . . . . . 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 4121	. . . . . 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 4348	. . . . . 6 ⊢ (∅ ∪ ( R ‘𝑥)) = ( R ‘𝑥)
107	105, 106	eqtrdi 2787	. . . . 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 7376	. . . 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 27900	. . . . 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 2775	. . 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 27941	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060   ∪ cun 3899  ∅c0 4285  {csn 4580  ‘cfv 6492  (class class class)co 7358   No csur 27607   |s ccuts 27755   0_s c0s 27801   1_s c1s 27802   L cleft 27821   R cright 27822   +_s cadds 27955   -_s csubs 28016   ·_s cmuls 28102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-muls 28103

This theorem is referenced by:  mulsridd  28110  mulslid  28138  n0seo  28417  zseo  28418  pw2recs  28434  addhalfcut  28455  pw2cutp1  28457  bdaypw2n0bndlem  28459  bdayfinbndlem1  28463  remulscllem1  28496