Theorem mulsrid 27963

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 7411	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (𝑥 ·s 1s ) = (𝑥𝑂 ·s 1s ))
2		id 22	. . 3 ⊢ (𝑥 = 𝑥𝑂 → 𝑥 = 𝑥𝑂)
3	1, 2	eqeq12d 2742	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝑥𝑂 ·s 1s ) = 𝑥𝑂))
4		oveq1 7411	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ·s 1s ) = (𝐴 ·s 1s ))
5		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
6	4, 5	eqeq12d 2742	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝐴 ·s 1s ) = 𝐴))
7		1sno 27710	. . . . . 6 ⊢ 1s ∈ No
8		mulsval 27959	. . . . . 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 688	. . . . 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 4171	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑝 ∈ ( L ‘𝑥) → 𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
12		oveq1 7411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑝 → (𝑥𝑂 ·s 1s ) = (𝑝 ·s 1s ))
13		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑝 → 𝑥𝑂 = 𝑝)
14	12, 13	eqeq12d 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑝 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑝 ·s 1s ) = 𝑝))
15	14	rspcva 3604	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑝 ·s 1s ) = 𝑝)
16	11, 15	sylan 579	. . . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 1s ) = 𝑝)
19		muls01 27962	. . . . . . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . 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		leftssno 27757	. . . . . . . . . . . . . . . . . 18 ⊢ ( L ‘𝑥) ⊆ No
24	23	sseli 3973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ ( L ‘𝑥) → 𝑝 ∈ No )
25	24	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → 𝑝 ∈ No )
26	25	addsridd 27832	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 +s 0s ) = 𝑝)
27	22, 26	eqtrd 2766	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → ((𝑝 ·s 1s ) +s (𝑥 ·s 0s )) = 𝑝)
28		muls01 27962	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ No → (𝑝 ·s 0s ) = 0s )
29	25, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 0s ) = 0s )
30	27, 29	oveq12d 7422	. . . . . . . . . . . . 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 ))
31		subsid1 27926	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ No → (𝑝 -s 0s ) = 𝑝)
32	25, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 -s 0s ) = 𝑝)
33	30, 32	eqtrd 2766	. . . . . . . . . . . 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)
34	33	eqeq2d 2737	. . . . . . . . . . 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))
35		equcom 2013	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑝 ↔ 𝑝 = 𝑎)
36	34, 35	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))
37	36	rexbidva 3170	. . . . . . . . 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))
38		left1s 27771	. . . . . . . . . . . 12 ⊢ ( L ‘ 1s ) = { 0s }
39	38	rexeqi 3318	. . . . . . . . . . 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)))
40		0sno 27709	. . . . . . . . . . . . 13 ⊢ 0s ∈ No
41	40	elexi 3488	. . . . . . . . . . . 12 ⊢ 0s ∈ V
42		oveq2 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑞 = 0s → (𝑥 ·s 𝑞) = (𝑥 ·s 0s ))
43	42	oveq2d 7420	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 0s → ((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) = ((𝑝 ·s 1s ) +s (𝑥 ·s 0s )))
44		oveq2 7412	. . . . . . . . . . . . . 14 ⊢ (𝑞 = 0s → (𝑝 ·s 𝑞) = (𝑝 ·s 0s ))
45	43, 44	oveq12d 7422	. . . . . . . . . . . . 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 )))
46	45	eqeq2d 2737	. . . . . . . . . . . 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 ))))
47	41, 46	rexsn 4681	. . . . . . . . . . 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 )))
48	39, 47	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 )))
49	48	rexbii 3088	. . . . . . . . 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 )))
50		risset 3224	. . . . . . . . 9 ⊢ (𝑎 ∈ ( L ‘𝑥) ↔ ∃𝑝 ∈ ( L ‘𝑥)𝑝 = 𝑎)
51	37, 49, 50	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)))
52	51	eqabcdv 2862	. . . . . . 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))
53		rex0 4352	. . . . . . . . . . . 12 ⊢ ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
54		right1s 27772	. . . . . . . . . . . . 13 ⊢ ( R ‘ 1s ) = ∅
55	54	rexeqi 3318	. . . . . . . . . . . 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)))
56	53, 55	mtbir 323	. . . . . . . . . . 11 ⊢ ¬ ∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
57	56	a1i 11	. . . . . . . . . 10 ⊢ (𝑟 ∈ ( R ‘𝑥) → ¬ ∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
58	57	nrex 3068	. . . . . . . . 9 ⊢ ¬ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
59	58	abf 4397	. . . . . . . 8 ⊢ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = ∅
60	59	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))} = ∅)
61	52, 60	uneq12d 4159	. . . . . 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) ∪ ∅))
62		un0 4385	. . . . . 6 ⊢ (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
63	61, 62	eqtrdi 2782	. . . . 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))
64		rex0 4352	. . . . . . . . . . . 12 ⊢ ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
65	54	rexeqi 3318	. . . . . . . . . . . 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)))
66	64, 65	mtbir 323	. . . . . . . . . . 11 ⊢ ¬ ∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
67	66	a1i 11	. . . . . . . . . 10 ⊢ (𝑡 ∈ ( L ‘𝑥) → ¬ ∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
68	67	nrex 3068	. . . . . . . . 9 ⊢ ¬ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
69	68	abf 4397	. . . . . . . 8 ⊢ {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = ∅
70	69	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))} = ∅)
71		elun2 4172	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑣 ∈ ( R ‘𝑥) → 𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
72		oveq1 7411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑣 → (𝑥𝑂 ·s 1s ) = (𝑣 ·s 1s ))
73		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥𝑂 = 𝑣 → 𝑥𝑂 = 𝑣)
74	72, 73	eqeq12d 2742	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥𝑂 = 𝑣 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑣 ·s 1s ) = 𝑣))
75	74	rspcva 3604	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑣 ·s 1s ) = 𝑣)
76	71, 75	sylan 579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ ( R ‘𝑥) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑣 ·s 1s ) = 𝑣)
77	76	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂 ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 1s ) = 𝑣)
78	77	adantll 711	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 1s ) = 𝑣)
79	20	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑥 ·s 0s ) = 0s )
80	78, 79	oveq12d 7422	. . . . . . . . . . . . . . 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 ))
81		rightssno 27758	. . . . . . . . . . . . . . . . . 18 ⊢ ( R ‘𝑥) ⊆ No
82	81	sseli 3973	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ ( R ‘𝑥) → 𝑣 ∈ No )
83	82	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → 𝑣 ∈ No )
84	83	addsridd 27832	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 +s 0s ) = 𝑣)
85	80, 84	eqtrd 2766	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → ((𝑣 ·s 1s ) +s (𝑥 ·s 0s )) = 𝑣)
86		muls01 27962	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ No → (𝑣 ·s 0s ) = 0s )
87	83, 86	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 0s ) = 0s )
88	85, 87	oveq12d 7422	. . . . . . . . . . . . 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 ))
89		subsid1 27926	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ No → (𝑣 -s 0s ) = 𝑣)
90	83, 89	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 -s 0s ) = 𝑣)
91	88, 90	eqtrd 2766	. . . . . . . . . . . 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)
92	91	eqeq2d 2737	. . . . . . . . . . 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))
93	38	rexeqi 3318	. . . . . . . . . . . 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)))
94		oveq2 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 0s → (𝑥 ·s 𝑤) = (𝑥 ·s 0s ))
95	94	oveq2d 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 0s → ((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) = ((𝑣 ·s 1s ) +s (𝑥 ·s 0s )))
96		oveq2 7412	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 0s → (𝑣 ·s 𝑤) = (𝑣 ·s 0s ))
97	95, 96	oveq12d 7422	. . . . . . . . . . . . . 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 )))
98	97	eqeq2d 2737	. . . . . . . . . . . . 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 ))))
99	41, 98	rexsn 4681	. . . . . . . . . . . 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 )))
100	93, 99	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 )))
101		equcom 2013	. . . . . . . . . . 11 ⊢ (𝑣 = 𝑑 ↔ 𝑑 = 𝑣)
102	92, 100, 101	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))
103	102	rexbidva 3170	. . . . . . . . 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))
104		risset 3224	. . . . . . . . 9 ⊢ (𝑑 ∈ ( R ‘𝑥) ↔ ∃𝑣 ∈ ( R ‘𝑥)𝑣 = 𝑑)
105	103, 104	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)))
106	105	eqabcdv 2862	. . . . . . 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))
107	70, 106	uneq12d 4159	. . . . . 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)))
108		0un 4387	. . . . . 6 ⊢ (∅ ∪ ( R ‘𝑥)) = ( R ‘𝑥)
109	107, 108	eqtrdi 2782	. . . . 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))
110	63, 109	oveq12d 7422	. . . 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)))
111		lrcut 27779	. . . . 5 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
112	111	adantr 480	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
113	10, 110, 112	3eqtrd 2770	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑥 ·s 1s ) = 𝑥)
114	113	ex 412	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂 → (𝑥 ·s 1s ) = 𝑥))
115	3, 6, 114	noinds 27812	1 ⊢ (𝐴 ∈ No → (𝐴 ·s 1s ) = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  ∀wral 3055  ∃wrex 3064   ∪ cun 3941  ∅c0 4317  {csn 4623  ‘cfv 6536  (class class class)co 7404   No csur 27523   |s cscut 27665   0_s c0s 27705   1_s c1s 27706   L cleft 27722   R cright 27723   +_s cadds 27826   -_s csubs 27883   ·_s cmuls 27956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-1o 8464  df-2o 8465  df-no 27526  df-slt 27527  df-bday 27528  df-sle 27628  df-sslt 27664  df-scut 27666  df-0s 27707  df-1s 27708  df-made 27724  df-old 27725  df-left 27727  df-right 27728  df-norec 27805  df-norec2 27816  df-adds 27827  df-negs 27884  df-subs 27885  df-muls 27957

This theorem is referenced by:  mulsridd  27964  mulslid  27992  remulscllem1  28178