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Theorem 0reno 28444
Description: Surreal zero is a surreal real. (Contributed by Scott Fenton, 15-Apr-2025.)
Assertion
Ref Expression
0reno 0s ∈ ℝs

Proof of Theorem 0reno
Dummy variables 𝑥 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0sno 27886 . 2 0s No
2 1nns 28367 . . . 4 1s ∈ ℕs
3 0slt1s 27889 . . . . . . 7 0s <s 1s
4 1sno 27887 . . . . . . . 8 1s No
5 sltneg 28092 . . . . . . . 8 (( 0s No ∧ 1s No ) → ( 0s <s 1s ↔ ( -us ‘ 1s ) <s ( -us ‘ 0s )))
61, 4, 5mp2an 692 . . . . . . 7 ( 0s <s 1s ↔ ( -us ‘ 1s ) <s ( -us ‘ 0s ))
73, 6mpbi 230 . . . . . 6 ( -us ‘ 1s ) <s ( -us ‘ 0s )
8 negs0s 28073 . . . . . 6 ( -us ‘ 0s ) = 0s
97, 8breqtri 5173 . . . . 5 ( -us ‘ 1s ) <s 0s
109, 3pm3.2i 470 . . . 4 (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )
11 fveq2 6907 . . . . . . 7 (𝑛 = 1s → ( -us𝑛) = ( -us ‘ 1s ))
1211breq1d 5158 . . . . . 6 (𝑛 = 1s → (( -us𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
13 breq2 5152 . . . . . 6 (𝑛 = 1s → ( 0s <s 𝑛 ↔ 0s <s 1s ))
1412, 13anbi12d 632 . . . . 5 (𝑛 = 1s → ((( -us𝑛) <s 0s ∧ 0s <s 𝑛) ↔ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )))
1514rspcev 3622 . . . 4 (( 1s ∈ ℕs ∧ (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )) → ∃𝑛 ∈ ℕs (( -us𝑛) <s 0s ∧ 0s <s 𝑛))
162, 10, 15mp2an 692 . . 3 𝑛 ∈ ℕs (( -us𝑛) <s 0s ∧ 0s <s 𝑛)
174a1i 11 . . . . . . . . . . 11 (𝑛 ∈ ℕs → 1s No )
18 nnsno 28344 . . . . . . . . . . 11 (𝑛 ∈ ℕs𝑛 No )
19 nnne0s 28355 . . . . . . . . . . 11 (𝑛 ∈ ℕs𝑛 ≠ 0s )
2017, 18, 19divscld 28263 . . . . . . . . . 10 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
2120negsval2d 28112 . . . . . . . . 9 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) = ( 0s -s ( 1s /su 𝑛)))
2221eqeq2d 2746 . . . . . . . 8 (𝑛 ∈ ℕs → (𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ 𝑥 = ( 0s -s ( 1s /su 𝑛))))
2322bicomd 223 . . . . . . 7 (𝑛 ∈ ℕs → (𝑥 = ( 0s -s ( 1s /su 𝑛)) ↔ 𝑥 = ( -us ‘( 1s /su 𝑛))))
2423rexbiia 3090 . . . . . 6 (∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)))
2524abbii 2807 . . . . 5 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))}
26 addslid 28016 . . . . . . . . 9 (( 1s /su 𝑛) ∈ No → ( 0s +s ( 1s /su 𝑛)) = ( 1s /su 𝑛))
2720, 26syl 17 . . . . . . . 8 (𝑛 ∈ ℕs → ( 0s +s ( 1s /su 𝑛)) = ( 1s /su 𝑛))
2827eqeq2d 2746 . . . . . . 7 (𝑛 ∈ ℕs → (𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ 𝑥 = ( 1s /su 𝑛)))
2928rexbiia 3090 . . . . . 6 (∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛))
3029abbii 2807 . . . . 5 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}
3125, 30oveq12i 7443 . . . 4 ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))}) = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)})
32 nnsex 28338 . . . . . . . . 9 s ∈ V
3332abrexex 7986 . . . . . . . 8 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∈ V
3433a1i 11 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∈ V)
35 snex 5442 . . . . . . . 8 { 0s } ∈ V
3635a1i 11 . . . . . . 7 (⊤ → { 0s } ∈ V)
3720negscld 28084 . . . . . . . . . . 11 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
38 eleq1 2827 . . . . . . . . . . 11 (𝑥 = ( -us ‘( 1s /su 𝑛)) → (𝑥 No ↔ ( -us ‘( 1s /su 𝑛)) ∈ No ))
3937, 38syl5ibrcom 247 . . . . . . . . . 10 (𝑛 ∈ ℕs → (𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 No ))
4039rexlimiv 3146 . . . . . . . . 9 (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 No )
4140a1i 11 . . . . . . . 8 (⊤ → (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 No ))
4241abssdv 4078 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ⊆ No )
431a1i 11 . . . . . . . 8 (⊤ → 0s No )
4443snssd 4814 . . . . . . 7 (⊤ → { 0s } ⊆ No )
45 vex 3482 . . . . . . . . . . . 12 𝑧 ∈ V
46 eqeq1 2739 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ 𝑧 = ( -us ‘( 1s /su 𝑛))))
4746rexbidv 3177 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛))))
4845, 47elab 3681 . . . . . . . . . . 11 (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)))
49 velsn 4647 . . . . . . . . . . 11 (𝑦 ∈ { 0s } ↔ 𝑦 = 0s )
5048, 49anbi12i 628 . . . . . . . . . 10 ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) ↔ (∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ))
51 r19.41v 3187 . . . . . . . . . 10 (∃𝑛 ∈ ℕs (𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) ↔ (∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ))
5250, 51bitr4i 278 . . . . . . . . 9 ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) ↔ ∃𝑛 ∈ ℕs (𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ))
53 muls02 28182 . . . . . . . . . . . . . . 15 (𝑛 No → ( 0s ·s 𝑛) = 0s )
5418, 53syl 17 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → ( 0s ·s 𝑛) = 0s )
5554, 3eqbrtrdi 5187 . . . . . . . . . . . . 13 (𝑛 ∈ ℕs → ( 0s ·s 𝑛) <s 1s )
561a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → 0s No )
57 nnsgt0 28357 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → 0s <s 𝑛)
5856, 17, 18, 57sltmuldivd 28268 . . . . . . . . . . . . 13 (𝑛 ∈ ℕs → (( 0s ·s 𝑛) <s 1s ↔ 0s <s ( 1s /su 𝑛)))
5955, 58mpbid 232 . . . . . . . . . . . 12 (𝑛 ∈ ℕs → 0s <s ( 1s /su 𝑛))
6020slt0neg2d 28098 . . . . . . . . . . . 12 (𝑛 ∈ ℕs → ( 0s <s ( 1s /su 𝑛) ↔ ( -us ‘( 1s /su 𝑛)) <s 0s ))
6159, 60mpbid 232 . . . . . . . . . . 11 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) <s 0s )
62 breq12 5153 . . . . . . . . . . 11 ((𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → (𝑧 <s 𝑦 ↔ ( -us ‘( 1s /su 𝑛)) <s 0s ))
6361, 62syl5ibrcom 247 . . . . . . . . . 10 (𝑛 ∈ ℕs → ((𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → 𝑧 <s 𝑦))
6463rexlimiv 3146 . . . . . . . . 9 (∃𝑛 ∈ ℕs (𝑧 = ( -us ‘( 1s /su 𝑛)) ∧ 𝑦 = 0s ) → 𝑧 <s 𝑦)
6552, 64sylbi 217 . . . . . . . 8 ((𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) → 𝑧 <s 𝑦)
66653adant1 1129 . . . . . . 7 ((⊤ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) → 𝑧 <s 𝑦)
6734, 36, 42, 44, 66ssltd 27851 . . . . . 6 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} <<s { 0s })
6832abrexex 7986 . . . . . . . 8 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V
6968a1i 11 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V)
70 eleq1 2827 . . . . . . . . . . 11 (𝑥 = ( 1s /su 𝑛) → (𝑥 No ↔ ( 1s /su 𝑛) ∈ No ))
7120, 70syl5ibrcom 247 . . . . . . . . . 10 (𝑛 ∈ ℕs → (𝑥 = ( 1s /su 𝑛) → 𝑥 No ))
7271rexlimiv 3146 . . . . . . . . 9 (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) → 𝑥 No )
7372a1i 11 . . . . . . . 8 (⊤ → (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) → 𝑥 No ))
7473abssdv 4078 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ⊆ No )
75 eqeq1 2739 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥 = ( 1s /su 𝑛) ↔ 𝑧 = ( 1s /su 𝑛)))
7675rexbidv 3177 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
7745, 76elab 3681 . . . . . . . . . . 11 (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ↔ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛))
7849, 77anbi12i 628 . . . . . . . . . 10 ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) ↔ (𝑦 = 0s ∧ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
79 r19.42v 3189 . . . . . . . . . 10 (∃𝑛 ∈ ℕs (𝑦 = 0s𝑧 = ( 1s /su 𝑛)) ↔ (𝑦 = 0s ∧ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
8078, 79bitr4i 278 . . . . . . . . 9 ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) ↔ ∃𝑛 ∈ ℕs (𝑦 = 0s𝑧 = ( 1s /su 𝑛)))
81 breq12 5153 . . . . . . . . . . . 12 ((𝑦 = 0s𝑧 = ( 1s /su 𝑛)) → (𝑦 <s 𝑧 ↔ 0s <s ( 1s /su 𝑛)))
8259, 81syl5ibrcom 247 . . . . . . . . . . 11 (𝑛 ∈ ℕs → ((𝑦 = 0s𝑧 = ( 1s /su 𝑛)) → 𝑦 <s 𝑧))
8382rexlimiv 3146 . . . . . . . . . 10 (∃𝑛 ∈ ℕs (𝑦 = 0s𝑧 = ( 1s /su 𝑛)) → 𝑦 <s 𝑧)
8483a1i 11 . . . . . . . . 9 (⊤ → (∃𝑛 ∈ ℕs (𝑦 = 0s𝑧 = ( 1s /su 𝑛)) → 𝑦 <s 𝑧))
8580, 84biimtrid 242 . . . . . . . 8 (⊤ → ((𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) → 𝑦 <s 𝑧))
86853impib 1115 . . . . . . 7 ((⊤ ∧ 𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) → 𝑦 <s 𝑧)
8736, 69, 44, 74, 86ssltd 27851 . . . . . 6 (⊤ → { 0s } <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)})
8867, 87cuteq0 27892 . . . . 5 (⊤ → ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s )
8988mptru 1544 . . . 4 ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s
9031, 89eqtr2i 2764 . . 3 0s = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))})
9116, 90pm3.2i 470 . 2 (∃𝑛 ∈ ℕs (( -us𝑛) <s 0s ∧ 0s <s 𝑛) ∧ 0s = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))}))
92 elreno 28442 . 2 ( 0s ∈ ℝs ↔ ( 0s No ∧ (∃𝑛 ∈ ℕs (( -us𝑛) <s 0s ∧ 0s <s 𝑛) ∧ 0s = ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛))}))))
931, 91, 92mpbir2an 711 1 0s ∈ ℝs
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wtru 1538  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  {csn 4631   class class class wbr 5148  cfv 6563  (class class class)co 7431   No csur 27699   <s cslt 27700   |s cscut 27842   0s c0s 27882   1s c1s 27883   +s cadds 28007   -us cnegs 28066   -s csubs 28067   ·s cmuls 28147   /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)
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