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Theorem renegscl 28239
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 27961 . . . 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 28209 . . . . . . . . . . . 12 (𝑛 ∈ ℕs𝑛 No )
43adantl 481 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → 𝑛 No )
54negscld 27962 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → ( -us𝑛) ∈ No )
6 simpl 482 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → 𝐴 No )
75, 6sltnegd 27972 . . . . . . . . 9 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝑛) <s 𝐴 ↔ ( -us𝐴) <s ( -us ‘( -us𝑛))))
8 negnegs 27969 . . . . . . . . . . 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 27972 . . . . . . . 8 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 <s 𝑛 ↔ ( -us𝑛) <s ( -us𝐴)))
1311, 12anbi12d 631 . . . . . . 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 3173 . . . . 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 716 . . 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 28237 . . . . . 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 772 . . . . 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 27980 . . . 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 27949 . . . . . . . . 9 -us Fn No
23 ssltss2 27735 . . . . . . . . . 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 6971 . . . . . . . . 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 2732 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2827rexbidv 3175 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2928rexab 3689 . . . . . . . . . 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 3282 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
31 ovex 7453 . . . . . . . . . . . . 13 (𝐴 +s ( 1s /su 𝑛)) ∈ V
32 fveqeq2 6906 . . . . . . . . . . . . 13 (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
3331, 32ceqsexv 3523 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
3433rexbii 3091 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
35 r19.41v 3185 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
3635exbii 1843 . . . . . . . . . . 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 27773 . . . . . . . . . . . . . . . . 17 1s No
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs → 1s No )
41 nnne0s 28218 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs𝑛 ≠ 0s )
4240, 3, 41divscld 28135 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
4342adantl 481 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
44 negsdi 27975 . . . . . . . . . . . . . 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 27984 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) -s ( 1s /su 𝑛)) = (( -us𝐴) +s ( -us ‘( 1s /su 𝑛))))
4845, 47eqtr4d 2771 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us𝐴) -s ( 1s /su 𝑛)))
4948eqeq1d 2730 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) -s ( 1s /su 𝑛)) = 𝑦))
50 eqcom 2735 . . . . . . . . . . 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 3173 . . . . . . . . 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 2863 . . . . . 6 (𝐴 No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))})
56 ssltss1 27734 . . . . . . . . . 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 6971 . . . . . . . . 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 2732 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6160rexbidv 3175 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6261rexab 3689 . . . . . . . . . 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 3282 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑧𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
64 ovex 7453 . . . . . . . . . . . . 13 (𝐴 -s ( 1s /su 𝑛)) ∈ V
65 fveqeq2 6906 . . . . . . . . . . . . 13 (𝑧 = (𝐴 -s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦))
6664, 65ceqsexv 3523 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
6766rexbii 3091 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
68 r19.41v 3185 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
6968exbii 1843 . . . . . . . . . . 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 27984 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
7372fveq2d 6901 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))))
7443negscld 27962 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
75 negsdi 27975 . . . . . . . . . . . . . 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 27969 . . . . . . . . . . . . . . 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 7436 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) +s ( -us ‘( -us ‘( 1s /su 𝑛)))) = (( -us𝐴) +s ( 1s /su 𝑛)))
8073, 76, 793eqtrd 2772 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = (( -us𝐴) +s ( 1s /su 𝑛)))
8180eqeq1d 2730 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82 eqcom 2735 . . . . . . . . . . 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 3173 . . . . . . . . 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 2863 . . . . . 6 (𝐴 No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) +s ( 1s /su 𝑛))})
8855, 87oveq12d 7438 . . . . 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 2768 . . 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 28236 . 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 28236 . 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 1534  wex 1774  wcel 2099  {cab 2705  wrex 3067  wss 3947   class class class wbr 5148  cima 5681   Fn wfn 6543  cfv 6548  (class class class)co 7420   No csur 27586   <s cslt 27587   <<s csslt 27726   |s cscut 27728   1s c1s 27769   +s cadds 27889   -us cnegs 27945   -s csubs 27946   /su cdivs 28100  scnns 28199  screno 28234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-dc 10470
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-nadd 8687  df-no 27589  df-slt 27590  df-bday 27591  df-sle 27691  df-sslt 27727  df-scut 27729  df-0s 27770  df-1s 27771  df-made 27787  df-old 27788  df-left 27790  df-right 27791  df-norec 27868  df-norec2 27879  df-adds 27890  df-negs 27947  df-subs 27948  df-muls 28020  df-divs 28101  df-n0s 28200  df-nns 28201  df-reno 28235
This theorem is referenced by: (None)
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