MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  renegscl Structured version   Visualization version   GIF version

Theorem renegscl 28445
Description: The surreal reals are closed under negation. Part of theorem 13(ii) of [Conway] p. 24. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
renegscl (𝐴 ∈ ℝs → ( -us𝐴) ∈ ℝs)

Proof of Theorem renegscl
Dummy variables 𝑛 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negscl 28083 . . . 4 (𝐴 No → ( -us𝐴) ∈ No )
21adantr 480 . . 3 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → ( -us𝐴) ∈ No )
3 nnsno 28344 . . . . . . . . . . . 12 (𝑛 ∈ ℕs𝑛 No )
43adantl 481 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → 𝑛 No )
54negscld 28084 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → ( -us𝑛) ∈ No )
6 simpl 482 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → 𝐴 No )
75, 6sltnegd 28094 . . . . . . . . 9 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝑛) <s 𝐴 ↔ ( -us𝐴) <s ( -us ‘( -us𝑛))))
8 negnegs 28091 . . . . . . . . . . 11 (𝑛 No → ( -us ‘( -us𝑛)) = 𝑛)
94, 8syl 17 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( -us𝑛)) = 𝑛)
109breq2d 5160 . . . . . . . . 9 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) <s ( -us ‘( -us𝑛)) ↔ ( -us𝐴) <s 𝑛))
117, 10bitrd 279 . . . . . . . 8 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝑛) <s 𝐴 ↔ ( -us𝐴) <s 𝑛))
126, 4sltnegd 28094 . . . . . . . 8 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 <s 𝑛 ↔ ( -us𝑛) <s ( -us𝐴)))
1311, 12anbi12d 632 . . . . . . 7 ((𝐴 No 𝑛 ∈ ℕs) → ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ↔ (( -us𝐴) <s 𝑛 ∧ ( -us𝑛) <s ( -us𝐴))))
1413biancomd 463 . . . . . 6 ((𝐴 No 𝑛 ∈ ℕs) → ((( -us𝑛) <s 𝐴𝐴 <s 𝑛) ↔ (( -us𝑛) <s ( -us𝐴) ∧ ( -us𝐴) <s 𝑛)))
1514rexbidva 3175 . . . . 5 (𝐴 No → (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ↔ ∃𝑛 ∈ ℕs (( -us𝑛) <s ( -us𝐴) ∧ ( -us𝐴) <s 𝑛)))
1615biimpa 476 . . . 4 ((𝐴 No ∧ ∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛)) → ∃𝑛 ∈ ℕs (( -us𝑛) <s ( -us𝐴) ∧ ( -us𝐴) <s 𝑛))
1716adantrr 717 . . 3 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → ∃𝑛 ∈ ℕs (( -us𝑛) <s ( -us𝐴) ∧ ( -us𝐴) <s 𝑛))
18 recut 28443 . . . . . 6 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
1918adantr 480 . . . . 5 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))})
20 simprr 773 . . . . 5 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))
2119, 20negsunif 28102 . . . 4 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → ( -us𝐴) = (( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})))
22 negsfn 28070 . . . . . . . . 9 -us Fn No
23 ssltss2 27849 . . . . . . . . . 10 ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
2418, 23syl 17 . . . . . . . . 9 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No )
25 fvelimab 6981 . . . . . . . . 9 (( -us Fn No ∧ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ⊆ No ) → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦))
2622, 24, 25sylancr 587 . . . . . . . 8 (𝐴 No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦))
27 eqeq1 2739 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2827rexbidv 3177 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2928rexab 3703 . . . . . . . . . 10 (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦 ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
30 rexcom4 3286 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
31 ovex 7464 . . . . . . . . . . . . 13 (𝐴 +s ( 1s /su 𝑛)) ∈ V
32 fveqeq2 6916 . . . . . . . . . . . . 13 (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
3331, 32ceqsexv 3530 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
3433rexbii 3092 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
35 r19.41v 3187 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
3635exbii 1845 . . . . . . . . . . 11 (∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
3730, 34, 363bitr3ri 302 . . . . . . . . . 10 (∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
3829, 37bitri 275 . . . . . . . . 9 (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
39 1sno 27887 . . . . . . . . . . . . . . . . 17 1s No
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs → 1s No )
41 nnne0s 28355 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs𝑛 ≠ 0s )
4240, 3, 41divscld 28263 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
4342adantl 481 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
44 negsdi 28097 . . . . . . . . . . . . . 14 ((𝐴 No ∧ ( 1s /su 𝑛) ∈ No ) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us𝐴) +s ( -us ‘( 1s /su 𝑛))))
4543, 44syldan 591 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us𝐴) +s ( -us ‘( 1s /su 𝑛))))
461adantr 480 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( -us𝐴) ∈ No )
4746, 43subsvald 28106 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) -s ( 1s /su 𝑛)) = (( -us𝐴) +s ( -us ‘( 1s /su 𝑛))))
4845, 47eqtr4d 2778 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us𝐴) -s ( 1s /su 𝑛)))
4948eqeq1d 2737 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) -s ( 1s /su 𝑛)) = 𝑦))
50 eqcom 2742 . . . . . . . . . . 11 ((( -us𝐴) -s ( 1s /su 𝑛)) = 𝑦𝑦 = (( -us𝐴) -s ( 1s /su 𝑛)))
5149, 50bitrdi 287 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))))
5251rexbidva 3175 . . . . . . . . 9 (𝐴 No → (∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))))
5338, 52bitrid 283 . . . . . . . 8 (𝐴 No → (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))))
5426, 53bitrd 279 . . . . . . 7 (𝐴 No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))))
5554eqabdv 2873 . . . . . 6 (𝐴 No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))})
56 ssltss1 27848 . . . . . . . . . 10 ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))} → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
5718, 56syl 17 . . . . . . . . 9 (𝐴 No → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No )
58 fvelimab 6981 . . . . . . . . 9 (( -us Fn No ∧ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ⊆ No ) → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦))
5922, 57, 58sylancr 587 . . . . . . . 8 (𝐴 No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) ↔ ∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦))
60 eqeq1 2739 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6160rexbidv 3177 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6261rexab 3703 . . . . . . . . . 10 (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦 ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
63 rexcom4 3286 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
64 ovex 7464 . . . . . . . . . . . . 13 (𝐴 -s ( 1s /su 𝑛)) ∈ V
65 fveqeq2 6916 . . . . . . . . . . . . 13 (𝑧 = (𝐴 -s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦))
6664, 65ceqsexv 3530 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
6766rexbii 3092 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
68 r19.41v 3187 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
6968exbii 1845 . . . . . . . . . . 11 (∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
7063, 67, 693bitr3ri 302 . . . . . . . . . 10 (∃𝑧(∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
7162, 70bitri 275 . . . . . . . . 9 (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
726, 43subsvald 28106 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
7372fveq2d 6911 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))))
7443negscld 28084 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
75 negsdi 28097 . . . . . . . . . . . . . 14 ((𝐴 No ∧ ( -us ‘( 1s /su 𝑛)) ∈ No ) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
7674, 75syldan 591 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
77 negnegs 28091 . . . . . . . . . . . . . . 15 (( 1s /su 𝑛) ∈ No → ( -us ‘( -us ‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
7843, 77syl 17 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( -us ‘( 1s /su 𝑛))) = ( 1s /su 𝑛))
7978oveq2d 7447 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))) = (( -us𝐴) +s ( 1s /su 𝑛)))
8073, 76, 793eqtrd 2779 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = (( -us𝐴) +s ( 1s /su 𝑛)))
8180eqeq1d 2737 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82 eqcom 2742 . . . . . . . . . . 11 ((( -us𝐴) +s ( 1s /su 𝑛)) = 𝑦𝑦 = (( -us𝐴) +s ( 1s /su 𝑛)))
8381, 82bitrdi 287 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))))
8483rexbidva 3175 . . . . . . . . 9 (𝐴 No → (∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))))
8571, 84bitrid 283 . . . . . . . 8 (𝐴 No → (∃𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} ( -us𝑧) = 𝑦 ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))))
8659, 85bitrd 279 . . . . . . 7 (𝐴 No → (𝑦 ∈ ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) ↔ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))))
8786eqabdv 2873 . . . . . 6 (𝐴 No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))})
8855, 87oveq12d 7449 . . . . 5 (𝐴 No → (( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))}))
8988adantr 480 . . . 4 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → (( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) |s ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))})) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))}))
9021, 89eqtrd 2775 . . 3 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → ( -us𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))}))
912, 17, 90jca32 515 . 2 ((𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))) → (( -us𝐴) ∈ No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s ( -us𝐴) ∧ ( -us𝐴) <s 𝑛) ∧ ( -us𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))}))))
92 elreno 28442 . 2 (𝐴 ∈ ℝs ↔ (𝐴 No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 𝐴𝐴 <s 𝑛) ∧ 𝐴 = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}))))
93 elreno 28442 . 2 (( -us𝐴) ∈ ℝs ↔ (( -us𝐴) ∈ No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s ( -us𝐴) ∧ ( -us𝐴) <s 𝑛) ∧ ( -us𝐴) = ({𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))} |s {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))}))))
9491, 92, 933imtr4i 292 1 (𝐴 ∈ ℝs → ( -us𝐴) ∈ ℝs)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068  wss 3963   class class class wbr 5148  cima 5692   Fn wfn 6558  cfv 6563  (class class class)co 7431   No csur 27699   <s cslt 27700   <<s csslt 27840   |s cscut 27842   1s c1s 27883   +s cadds 28007   -us cnegs 28066   -s csubs 28067   /su cdivs 28228  scnns 28334  screno 28440
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-reno 28441
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator