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Theorem renegscl 28167
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 27889 . . . 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 28137 . . . . . . . . . . . 12 (𝑛 ∈ ℕs𝑛 No )
43adantl 481 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → 𝑛 No )
54negscld 27890 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → ( -us𝑛) ∈ No )
6 simpl 482 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → 𝐴 No )
75, 6sltnegd 27900 . . . . . . . . 9 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝑛) <s 𝐴 ↔ ( -us𝐴) <s ( -us ‘( -us𝑛))))
8 negnegs 27897 . . . . . . . . . . 11 (𝑛 No → ( -us ‘( -us𝑛)) = 𝑛)
94, 8syl 17 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( -us𝑛)) = 𝑛)
109breq2d 5151 . . . . . . . . 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 27900 . . . . . . . 8 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 <s 𝑛 ↔ ( -us𝑛) <s ( -us𝐴)))
1311, 12anbi12d 630 . . . . . . 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 3168 . . . . 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 714 . . 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 28165 . . . . . 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 770 . . . . 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 27908 . . . 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 27877 . . . . . . . . 9 -us Fn No
23 ssltss2 27663 . . . . . . . . . 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 6955 . . . . . . . . 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 586 . . . . . . . 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 2728 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2827rexbidv 3170 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2928rexab 3683 . . . . . . . . . 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 3277 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
31 ovex 7435 . . . . . . . . . . . . 13 (𝐴 +s ( 1s /su 𝑛)) ∈ V
32 fveqeq2 6891 . . . . . . . . . . . . 13 (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
3331, 32ceqsexv 3518 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
3433rexbii 3086 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
35 r19.41v 3180 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
3635exbii 1842 . . . . . . . . . . 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 27701 . . . . . . . . . . . . . . . . 17 1s No
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs → 1s No )
41 nnne0s 28146 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs𝑛 ≠ 0s )
4240, 3, 41divscld 28063 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
4342adantl 481 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
44 negsdi 27903 . . . . . . . . . . . . . 14 ((𝐴 No ∧ ( 1s /su 𝑛) ∈ No ) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us𝐴) +s ( -us ‘( 1s /su 𝑛))))
4543, 44syldan 590 . . . . . . . . . . . . 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 27912 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) -s ( 1s /su 𝑛)) = (( -us𝐴) +s ( -us ‘( 1s /su 𝑛))))
4845, 47eqtr4d 2767 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us𝐴) -s ( 1s /su 𝑛)))
4948eqeq1d 2726 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) -s ( 1s /su 𝑛)) = 𝑦))
50 eqcom 2731 . . . . . . . . . . 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 3168 . . . . . . . . 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 2859 . . . . . 6 (𝐴 No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))})
56 ssltss1 27662 . . . . . . . . . 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 6955 . . . . . . . . 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 586 . . . . . . . 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 2728 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6160rexbidv 3170 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6261rexab 3683 . . . . . . . . . 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 3277 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
64 ovex 7435 . . . . . . . . . . . . 13 (𝐴 -s ( 1s /su 𝑛)) ∈ V
65 fveqeq2 6891 . . . . . . . . . . . . 13 (𝑧 = (𝐴 -s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦))
6664, 65ceqsexv 3518 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
6766rexbii 3086 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
68 r19.41v 3180 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
6968exbii 1842 . . . . . . . . . . 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 27912 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
7372fveq2d 6886 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))))
7443negscld 27890 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
75 negsdi 27903 . . . . . . . . . . . . . 14 ((𝐴 No ∧ ( -us ‘( 1s /su 𝑛)) ∈ No ) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
7674, 75syldan 590 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))) = (( -us𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))))
77 negnegs 27897 . . . . . . . . . . . . . . 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 7418 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))) = (( -us𝐴) +s ( 1s /su 𝑛)))
8073, 76, 793eqtrd 2768 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = (( -us𝐴) +s ( 1s /su 𝑛)))
8180eqeq1d 2726 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82 eqcom 2731 . . . . . . . . . . 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 3168 . . . . . . . . 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 2859 . . . . . 6 (𝐴 No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))})
8855, 87oveq12d 7420 . . . . 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 2764 . . 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 28164 . 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 28164 . 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 205  wa 395   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wrex 3062  wss 3941   class class class wbr 5139  cima 5670   Fn wfn 6529  cfv 6534  (class class class)co 7402   No csur 27514   <s cslt 27515   <<s csslt 27654   |s cscut 27656   1s c1s 27697   +s cadds 27817   -us cnegs 27873   -s csubs 27874   /su cdivs 28028  scnns 28127  screno 28162
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 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-dc 10438
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-ot 4630  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-oadd 8466  df-nadd 8662  df-no 27517  df-slt 27518  df-bday 27519  df-sle 27619  df-sslt 27655  df-scut 27657  df-0s 27698  df-1s 27699  df-made 27715  df-old 27716  df-left 27718  df-right 27719  df-norec 27796  df-norec2 27807  df-adds 27818  df-negs 27875  df-subs 27876  df-muls 27948  df-divs 28029  df-n0s 28128  df-nns 28129  df-reno 28163
This theorem is referenced by: (None)
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