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Theorem 0reno 28429
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 27871 . 2 0s No
2 1nns 28352 . . . 4 1s ∈ ℕs
3 0slt1s 27874 . . . . . . 7 0s <s 1s
4 1sno 27872 . . . . . . . 8 1s No
5 sltneg 28077 . . . . . . . 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 28058 . . . . . 6 ( -us ‘ 0s ) = 0s
97, 8breqtri 5168 . . . . 5 ( -us ‘ 1s ) <s 0s
109, 3pm3.2i 470 . . . 4 (( -us ‘ 1s ) <s 0s ∧ 0s <s 1s )
11 fveq2 6906 . . . . . . 7 (𝑛 = 1s → ( -us𝑛) = ( -us ‘ 1s ))
1211breq1d 5153 . . . . . 6 (𝑛 = 1s → (( -us𝑛) <s 0s ↔ ( -us ‘ 1s ) <s 0s ))
13 breq2 5147 . . . . . 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 28329 . . . . . . . . . . 11 (𝑛 ∈ ℕs𝑛 No )
19 nnne0s 28340 . . . . . . . . . . 11 (𝑛 ∈ ℕs𝑛 ≠ 0s )
2017, 18, 19divscld 28248 . . . . . . . . . 10 (𝑛 ∈ ℕs → ( 1s /su 𝑛) ∈ No )
2120negsval2d 28097 . . . . . . . . 9 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) = ( 0s -s ( 1s /su 𝑛)))
2221eqeq2d 2748 . . . . . . . 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 3092 . . . . . 6 (∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)))
2524abbii 2809 . . . . 5 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 0s -s ( 1s /su 𝑛))} = {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))}
26 addslid 28001 . . . . . . . . 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 2748 . . . . . . 7 (𝑛 ∈ ℕs → (𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ 𝑥 = ( 1s /su 𝑛)))
2928rexbiia 3092 . . . . . 6 (∃𝑛 ∈ ℕs 𝑥 = ( 0s +s ( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛))
3029abbii 2809 . . . . 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 28323 . . . . . . . . 9 s ∈ V
3332abrexex 7987 . . . . . . . 8 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∈ V
3433a1i 11 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∈ V)
35 snex 5436 . . . . . . . 8 { 0s } ∈ V
3635a1i 11 . . . . . . 7 (⊤ → { 0s } ∈ V)
3720negscld 28069 . . . . . . . . . . 11 (𝑛 ∈ ℕs → ( -us ‘( 1s /su 𝑛)) ∈ No )
38 eleq1 2829 . . . . . . . . . . 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 3148 . . . . . . . . 9 (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 No )
4140a1i 11 . . . . . . . 8 (⊤ → (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) → 𝑥 No ))
4241abssdv 4068 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ⊆ No )
431a1i 11 . . . . . . . 8 (⊤ → 0s No )
4443snssd 4809 . . . . . . 7 (⊤ → { 0s } ⊆ No )
45 vex 3484 . . . . . . . . . . . 12 𝑧 ∈ V
46 eqeq1 2741 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ 𝑧 = ( -us ‘( 1s /su 𝑛))))
4746rexbidv 3179 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛)) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛))))
4845, 47elab 3679 . . . . . . . . . . 11 (𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ↔ ∃𝑛 ∈ ℕs 𝑧 = ( -us ‘( 1s /su 𝑛)))
49 velsn 4642 . . . . . . . . . . 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 3189 . . . . . . . . . 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 28167 . . . . . . . . . . . . . . 15 (𝑛 No → ( 0s ·s 𝑛) = 0s )
5418, 53syl 17 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → ( 0s ·s 𝑛) = 0s )
5554, 3eqbrtrdi 5182 . . . . . . . . . . . . 13 (𝑛 ∈ ℕs → ( 0s ·s 𝑛) <s 1s )
561a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → 0s No )
57 nnsgt0 28342 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕs → 0s <s 𝑛)
5856, 17, 18, 57sltmuldivd 28253 . . . . . . . . . . . . 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 28083 . . . . . . . . . . . 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 5148 . . . . . . . . . . 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 3148 . . . . . . . . 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 1131 . . . . . . 7 ((⊤ ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} ∧ 𝑦 ∈ { 0s }) → 𝑧 <s 𝑦)
6734, 36, 42, 44, 66ssltd 27836 . . . . . 6 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} <<s { 0s })
6832abrexex 7987 . . . . . . . 8 {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V
6968a1i 11 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ∈ V)
70 eleq1 2829 . . . . . . . . . . 11 (𝑥 = ( 1s /su 𝑛) → (𝑥 No ↔ ( 1s /su 𝑛) ∈ No ))
7120, 70syl5ibrcom 247 . . . . . . . . . 10 (𝑛 ∈ ℕs → (𝑥 = ( 1s /su 𝑛) → 𝑥 No ))
7271rexlimiv 3148 . . . . . . . . 9 (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) → 𝑥 No )
7372a1i 11 . . . . . . . 8 (⊤ → (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) → 𝑥 No ))
7473abssdv 4068 . . . . . . 7 (⊤ → {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)} ⊆ No )
75 eqeq1 2741 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥 = ( 1s /su 𝑛) ↔ 𝑧 = ( 1s /su 𝑛)))
7675rexbidv 3179 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛) ↔ ∃𝑛 ∈ ℕs 𝑧 = ( 1s /su 𝑛)))
7745, 76elab 3679 . . . . . . . . . . 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 3191 . . . . . . . . . 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 5148 . . . . . . . . . . . 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 3148 . . . . . . . . . 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 1117 . . . . . . 7 ((⊤ ∧ 𝑦 ∈ { 0s } ∧ 𝑧 ∈ {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) → 𝑦 <s 𝑧)
8736, 69, 44, 74, 86ssltd 27836 . . . . . 6 (⊤ → { 0s } <<s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)})
8867, 87cuteq0 27877 . . . . 5 (⊤ → ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s )
8988mptru 1547 . . . 4 ({𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( -us ‘( 1s /su 𝑛))} |s {𝑥 ∣ ∃𝑛 ∈ ℕs 𝑥 = ( 1s /su 𝑛)}) = 0s
9031, 89eqtr2i 2766 . . 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 28427 . 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 1540  wtru 1541  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  {csn 4626   class class class wbr 5143  cfv 6561  (class class class)co 7431   No csur 27684   <s cslt 27685   |s cscut 27827   0s c0s 27867   1s c1s 27868   +s cadds 27992   -us cnegs 28051   -s csubs 28052   ·s cmuls 28132   /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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