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Theorem renegscl 28430
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 28068 . . . 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 28329 . . . . . . . . . . . 12 (𝑛 ∈ ℕs𝑛 No )
43adantl 481 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → 𝑛 No )
54negscld 28069 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → ( -us𝑛) ∈ No )
6 simpl 482 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → 𝐴 No )
75, 6sltnegd 28079 . . . . . . . . 9 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝑛) <s 𝐴 ↔ ( -us𝐴) <s ( -us ‘( -us𝑛))))
8 negnegs 28076 . . . . . . . . . . 11 (𝑛 No → ( -us ‘( -us𝑛)) = 𝑛)
94, 8syl 17 . . . . . . . . . 10 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( -us𝑛)) = 𝑛)
109breq2d 5155 . . . . . . . . 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 28079 . . . . . . . 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 3177 . . . . 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 28428 . . . . . 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 28087 . . . 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 28055 . . . . . . . . 9 -us Fn No
23 ssltss2 27834 . . . . . . . . . 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 2741 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2827rexbidv 3179 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛))))
2928rexab 3700 . . . . . . . . . 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 3288 . . . . . . . . . . 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 6915 . . . . . . . . . . . . 13 (𝑧 = (𝐴 +s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦))
3331, 32ceqsexv 3532 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
3433rexbii 3094 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦)
35 r19.41v 3189 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 +s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
3635exbii 1848 . . . . . . . . . . 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 27872 . . . . . . . . . . . . . . . . 17 1s No
4039a1i 11 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs → 1s No )
41 nnne0s 28340 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕs𝑛 ≠ 0s )
4240, 3, 41divscld 28248 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
4342adantl 481 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( 1s /su 𝑛) ∈ No )
44 negsdi 28082 . . . . . . . . . . . . . 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 28091 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → (( -us𝐴) -s ( 1s /su 𝑛)) = (( -us𝐴) +s ( -us ‘( 1s /su 𝑛))))
4845, 47eqtr4d 2780 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 +s ( 1s /su 𝑛))) = (( -us𝐴) -s ( 1s /su 𝑛)))
4948eqeq1d 2739 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 +s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) -s ( 1s /su 𝑛)) = 𝑦))
50 eqcom 2744 . . . . . . . . . . 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 3177 . . . . . . . . 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 2875 . . . . . 6 (𝐴 No → ( -us “ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = (𝐴 +s ( 1s /su 𝑛))}) = {𝑦 ∣ ∃𝑛 ∈ ℕs 𝑦 = (( -us𝐴) -s ( 1s /su 𝑛))})
56 ssltss1 27833 . . . . . . . . . 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 2741 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6160rexbidv 3179 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = (𝐴 -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛))))
6261rexab 3700 . . . . . . . . . 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 3288 . . . . . . . . . . 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 6915 . . . . . . . . . . . . 13 (𝑧 = (𝐴 -s ( 1s /su 𝑛)) → (( -us𝑧) = 𝑦 ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦))
6664, 65ceqsexv 3532 . . . . . . . . . . . 12 (∃𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
6766rexbii 3094 . . . . . . . . . . 11 (∃𝑛 ∈ ℕs𝑧(𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ ∃𝑛 ∈ ℕs ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦)
68 r19.41v 3189 . . . . . . . . . . . 12 (∃𝑛 ∈ ℕs (𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦) ↔ (∃𝑛 ∈ ℕs 𝑧 = (𝐴 -s ( 1s /su 𝑛)) ∧ ( -us𝑧) = 𝑦))
6968exbii 1848 . . . . . . . . . . 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 28091 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → (𝐴 -s ( 1s /su 𝑛)) = (𝐴 +s ( -us ‘( 1s /su 𝑛))))
7372fveq2d 6910 . . . . . . . . . . . . 13 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = ( -us ‘(𝐴 +s ( -us ‘( 1s /su 𝑛)))))
7443negscld 28069 . . . . . . . . . . . . . 14 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘( 1s /su 𝑛)) ∈ No )
75 negsdi 28082 . . . . . . . . . . . . . 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 28076 . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . 12 ((𝐴 No 𝑛 ∈ ℕs) → ( -us ‘(𝐴 -s ( 1s /su 𝑛))) = (( -us𝐴) +s ( 1s /su 𝑛)))
8180eqeq1d 2739 . . . . . . . . . . 11 ((𝐴 No 𝑛 ∈ ℕs) → (( -us ‘(𝐴 -s ( 1s /su 𝑛))) = 𝑦 ↔ (( -us𝐴) +s ( 1s /su 𝑛)) = 𝑦))
82 eqcom 2744 . . . . . . . . . . 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 3177 . . . . . . . . 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 2875 . . . . . 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 2777 . . 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 28427 . 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 28427 . 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 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  wss 3951   class class class wbr 5143  cima 5688   Fn wfn 6556  cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   <<s csslt 27825   |s cscut 27827   1s c1s 27868   +s cadds 27992   -us cnegs 28051   -s csubs 28052   /su cdivs 28213  scnns 28319  screno 28425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-dc 10486
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053  df-subs 28054  df-muls 28133  df-divs 28214  df-n0s 28320  df-nns 28321  df-reno 28426
This theorem is referenced by: (None)
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