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Theorem mulsrid 28154
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 𝑥𝑂 𝑎 𝑏 𝑐 𝑑 𝑝 𝑞 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . 3 (𝑥 = 𝑥𝑂 → (𝑥 ·s 1s ) = (𝑥𝑂 ·s 1s ))
2 id 22 . . 3 (𝑥 = 𝑥𝑂𝑥 = 𝑥𝑂)
31, 2eqeq12d 2751 . 2 (𝑥 = 𝑥𝑂 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝑥𝑂 ·s 1s ) = 𝑥𝑂))
4 oveq1 7438 . . 3 (𝑥 = 𝐴 → (𝑥 ·s 1s ) = (𝐴 ·s 1s ))
5 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
64, 5eqeq12d 2751 . 2 (𝑥 = 𝐴 → ((𝑥 ·s 1s ) = 𝑥 ↔ (𝐴 ·s 1s ) = 𝐴))
7 1sno 27887 . . . . . 6 1s No
8 mulsval 28150 . . . . . 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))})))
97, 8mpan2 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))})))
109adantr 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 4192 . . . . . . . . . . . . . . . . . . 19 (𝑝 ∈ ( L ‘𝑥) → 𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
12 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑂 = 𝑝 → (𝑥𝑂 ·s 1s ) = (𝑝 ·s 1s ))
13 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑂 = 𝑝𝑥𝑂 = 𝑝)
1412, 13eqeq12d 2751 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑂 = 𝑝 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑝 ·s 1s ) = 𝑝))
1514rspcva 3620 . . . . . . . . . . . . . . . . . . 19 ((𝑝 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑝 ·s 1s ) = 𝑝)
1611, 15sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝑝 ∈ ( L ‘𝑥) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑝 ·s 1s ) = 𝑝)
1716ancoms 458 . . . . . . . . . . . . . . . . 17 ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 1s ) = 𝑝)
1817adantll 714 . . . . . . . . . . . . . . . 16 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 1s ) = 𝑝)
19 muls01 28153 . . . . . . . . . . . . . . . . . 18 (𝑥 No → (𝑥 ·s 0s ) = 0s )
2019adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑥 ·s 0s ) = 0s )
2120adantr 480 . . . . . . . . . . . . . . . 16 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑥 ·s 0s ) = 0s )
2218, 21oveq12d 7449 . . . . . . . . . . . . . . 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 27934 . . . . . . . . . . . . . . . . . 18 ( L ‘𝑥) ⊆ No
2423sseli 3991 . . . . . . . . . . . . . . . . 17 (𝑝 ∈ ( L ‘𝑥) → 𝑝 No )
2524adantl 481 . . . . . . . . . . . . . . . 16 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → 𝑝 No )
2625addsridd 28013 . . . . . . . . . . . . . . 15 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 +s 0s ) = 𝑝)
2722, 26eqtrd 2775 . . . . . . . . . . . . . 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 28153 . . . . . . . . . . . . . . 15 (𝑝 No → (𝑝 ·s 0s ) = 0s )
2925, 28syl 17 . . . . . . . . . . . . . 14 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 ·s 0s ) = 0s )
3027, 29oveq12d 7449 . . . . . . . . . . . . 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 28113 . . . . . . . . . . . . . 14 (𝑝 No → (𝑝 -s 0s ) = 𝑝)
3225, 31syl 17 . . . . . . . . . . . . 13 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑝 ∈ ( L ‘𝑥)) → (𝑝 -s 0s ) = 𝑝)
3330, 32eqtrd 2775 . . . . . . . . . . . 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)
3433eqeq2d 2746 . . . . . . . . . . 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 2015 . . . . . . . . . . 11 (𝑎 = 𝑝𝑝 = 𝑎)
3634, 35bitrdi 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))
3736rexbidva 3175 . . . . . . . . 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 27948 . . . . . . . . . . . 12 ( L ‘ 1s ) = { 0s }
3938rexeqi 3323 . . . . . . . . . . 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 27886 . . . . . . . . . . . . 13 0s No
4140elexi 3501 . . . . . . . . . . . 12 0s ∈ V
42 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑞 = 0s → (𝑥 ·s 𝑞) = (𝑥 ·s 0s ))
4342oveq2d 7447 . . . . . . . . . . . . . 14 (𝑞 = 0s → ((𝑝 ·s 1s ) +s (𝑥 ·s 𝑞)) = ((𝑝 ·s 1s ) +s (𝑥 ·s 0s )))
44 oveq2 7439 . . . . . . . . . . . . . 14 (𝑞 = 0s → (𝑝 ·s 𝑞) = (𝑝 ·s 0s ))
4543, 44oveq12d 7449 . . . . . . . . . . . . 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 )))
4645eqeq2d 2746 . . . . . . . . . . . 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 ))))
4741, 46rexsn 4687 . . . . . . . . . . 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 )))
4839, 47bitri 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 )))
4948rexbii 3092 . . . . . . . . 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 3231 . . . . . . . . 9 (𝑎 ∈ ( L ‘𝑥) ↔ ∃𝑝 ∈ ( L ‘𝑥)𝑝 = 𝑎)
5137, 49, 503bitr4g 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)))
5251eqabcdv 2874 . . . . . . 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 4366 . . . . . . . . . . . 12 ¬ ∃𝑠 ∈ ∅ 𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
54 right1s 27949 . . . . . . . . . . . . 13 ( R ‘ 1s ) = ∅
5554rexeqi 3323 . . . . . . . . . . . 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)))
5653, 55mtbir 323 . . . . . . . . . . 11 ¬ ∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
5756a1i 11 . . . . . . . . . 10 (𝑟 ∈ ( R ‘𝑥) → ¬ ∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
5857nrex 3072 . . . . . . . . 9 ¬ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))
5958abf 4412 . . . . . . . 8 {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝑥)∃𝑠 ∈ ( R ‘ 1s )𝑏 = (((𝑟 ·s 1s ) +s (𝑥 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = ∅
6059a1i 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))} = ∅)
6152, 60uneq12d 4179 . . . . . 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 4400 . . . . . 6 (( L ‘𝑥) ∪ ∅) = ( L ‘𝑥)
6361, 62eqtrdi 2791 . . . . 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 4366 . . . . . . . . . . . 12 ¬ ∃𝑢 ∈ ∅ 𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
6554rexeqi 3323 . . . . . . . . . . . 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)))
6664, 65mtbir 323 . . . . . . . . . . 11 ¬ ∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
6766a1i 11 . . . . . . . . . 10 (𝑡 ∈ ( L ‘𝑥) → ¬ ∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
6867nrex 3072 . . . . . . . . 9 ¬ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))
6968abf 4412 . . . . . . . 8 {𝑐 ∣ ∃𝑡 ∈ ( L ‘𝑥)∃𝑢 ∈ ( R ‘ 1s )𝑐 = (((𝑡 ·s 1s ) +s (𝑥 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = ∅
7069a1i 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 4193 . . . . . . . . . . . . . . . . . . 19 (𝑣 ∈ ( R ‘𝑥) → 𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)))
72 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑂 = 𝑣 → (𝑥𝑂 ·s 1s ) = (𝑣 ·s 1s ))
73 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑂 = 𝑣𝑥𝑂 = 𝑣)
7472, 73eqeq12d 2751 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑂 = 𝑣 → ((𝑥𝑂 ·s 1s ) = 𝑥𝑂 ↔ (𝑣 ·s 1s ) = 𝑣))
7574rspcva 3620 . . . . . . . . . . . . . . . . . . 19 ((𝑣 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥)) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑣 ·s 1s ) = 𝑣)
7671, 75sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝑣 ∈ ( R ‘𝑥) ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑣 ·s 1s ) = 𝑣)
7776ancoms 458 . . . . . . . . . . . . . . . . 17 ((∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 1s ) = 𝑣)
7877adantll 714 . . . . . . . . . . . . . . . 16 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 1s ) = 𝑣)
7920adantr 480 . . . . . . . . . . . . . . . 16 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑥 ·s 0s ) = 0s )
8078, 79oveq12d 7449 . . . . . . . . . . . . . . 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 27935 . . . . . . . . . . . . . . . . . 18 ( R ‘𝑥) ⊆ No
8281sseli 3991 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ ( R ‘𝑥) → 𝑣 No )
8382adantl 481 . . . . . . . . . . . . . . . 16 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → 𝑣 No )
8483addsridd 28013 . . . . . . . . . . . . . . 15 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 +s 0s ) = 𝑣)
8580, 84eqtrd 2775 . . . . . . . . . . . . . 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 28153 . . . . . . . . . . . . . . 15 (𝑣 No → (𝑣 ·s 0s ) = 0s )
8783, 86syl 17 . . . . . . . . . . . . . 14 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 ·s 0s ) = 0s )
8885, 87oveq12d 7449 . . . . . . . . . . . . 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 28113 . . . . . . . . . . . . . 14 (𝑣 No → (𝑣 -s 0s ) = 𝑣)
9083, 89syl 17 . . . . . . . . . . . . 13 (((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) ∧ 𝑣 ∈ ( R ‘𝑥)) → (𝑣 -s 0s ) = 𝑣)
9188, 90eqtrd 2775 . . . . . . . . . . . 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)
9291eqeq2d 2746 . . . . . . . . . . 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))
9338rexeqi 3323 . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . 16 (𝑤 = 0s → (𝑥 ·s 𝑤) = (𝑥 ·s 0s ))
9594oveq2d 7447 . . . . . . . . . . . . . . 15 (𝑤 = 0s → ((𝑣 ·s 1s ) +s (𝑥 ·s 𝑤)) = ((𝑣 ·s 1s ) +s (𝑥 ·s 0s )))
96 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑤 = 0s → (𝑣 ·s 𝑤) = (𝑣 ·s 0s ))
9795, 96oveq12d 7449 . . . . . . . . . . . . . 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 )))
9897eqeq2d 2746 . . . . . . . . . . . . 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 ))))
9941, 98rexsn 4687 . . . . . . . . . . . 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 )))
10093, 99bitri 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 2015 . . . . . . . . . . 11 (𝑣 = 𝑑𝑑 = 𝑣)
10292, 100, 1013bitr4g 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))
103102rexbidva 3175 . . . . . . . . 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 3231 . . . . . . . . 9 (𝑑 ∈ ( R ‘𝑥) ↔ ∃𝑣 ∈ ( R ‘𝑥)𝑣 = 𝑑)
105103, 104bitr4di 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)))
106105eqabcdv 2874 . . . . . . 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))
10770, 106uneq12d 4179 . . . . . 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 4402 . . . . . 6 (∅ ∪ ( R ‘𝑥)) = ( R ‘𝑥)
109107, 108eqtrdi 2791 . . . . 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))
11063, 109oveq12d 7449 . . . 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 27956 . . . . 5 (𝑥 No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
112111adantr 480 . . . 4 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
11310, 110, 1123eqtrd 2779 . . 3 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂) → (𝑥 ·s 1s ) = 𝑥)
114113ex 412 . 2 (𝑥 No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))(𝑥𝑂 ·s 1s ) = 𝑥𝑂 → (𝑥 ·s 1s ) = 𝑥))
1153, 6, 114noinds 27993 1 (𝐴 No → (𝐴 ·s 1s ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  cun 3961  c0 4339  {csn 4631  cfv 6563  (class class class)co 7431   No csur 27699   |s cscut 27842   0s c0s 27882   1s c1s 27883   L cleft 27899   R cright 27900   +s cadds 28007   -s csubs 28067   ·s cmuls 28147
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
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-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-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-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  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
This theorem is referenced by:  mulsridd  28155  mulslid  28183  n0seo  28420  zseo  28421  addhalfcut  28434  zs12bday  28439  remulscllem1  28447
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