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Theorem n0seo 28414
Description: A non-negative surreal integer is either even or odd. (Contributed by Scott Fenton, 19-Aug-2025.)
Assertion
Ref Expression
n0seo (𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))
Distinct variable group:   𝑥,𝑁

Proof of Theorem n0seo
Dummy variables 𝑛 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2738 . . . 4 (𝑚 = 0s → (𝑚 = (2s ·s 𝑥) ↔ 0s = (2s ·s 𝑥)))
21rexbidv 3181 . . 3 (𝑚 = 0s → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥)))
3 eqeq1 2738 . . . 4 (𝑚 = 0s → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 0s = ((2s ·s 𝑥) +s 1s )))
43rexbidv 3181 . . 3 (𝑚 = 0s → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s )))
52, 4orbi12d 917 . 2 (𝑚 = 0s → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s ))))
6 eqeq1 2738 . . . 4 (𝑚 = 𝑛 → (𝑚 = (2s ·s 𝑥) ↔ 𝑛 = (2s ·s 𝑥)))
76rexbidv 3181 . . 3 (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥)))
8 eqeq1 2738 . . . 4 (𝑚 = 𝑛 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑛 = ((2s ·s 𝑥) +s 1s )))
98rexbidv 3181 . . 3 (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )))
107, 9orbi12d 917 . 2 (𝑚 = 𝑛 → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s ))))
11 eqeq1 2738 . . . . 5 (𝑚 = (𝑛 +s 1s ) → (𝑚 = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑥)))
1211rexbidv 3181 . . . 4 (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑥)))
13 oveq2 7453 . . . . . 6 (𝑥 = 𝑦 → (2s ·s 𝑥) = (2s ·s 𝑦))
1413eqeq2d 2745 . . . . 5 (𝑥 = 𝑦 → ((𝑛 +s 1s ) = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑦)))
1514cbvrexvw 3239 . . . 4 (∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑥) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦))
1612, 15bitrdi 287 . . 3 (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
17 eqeq1 2738 . . . . 5 (𝑚 = (𝑛 +s 1s ) → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s )))
1817rexbidv 3181 . . . 4 (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s )))
1913oveq1d 7460 . . . . . 6 (𝑥 = 𝑦 → ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
2019eqeq2d 2745 . . . . 5 (𝑥 = 𝑦 → ((𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
2120cbvrexvw 3239 . . . 4 (∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))
2218, 21bitrdi 287 . . 3 (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
2316, 22orbi12d 917 . 2 (𝑚 = (𝑛 +s 1s ) → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))))
24 eqeq1 2738 . . . 4 (𝑚 = 𝑁 → (𝑚 = (2s ·s 𝑥) ↔ 𝑁 = (2s ·s 𝑥)))
2524rexbidv 3181 . . 3 (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥)))
26 eqeq1 2738 . . . 4 (𝑚 = 𝑁 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑁 = ((2s ·s 𝑥) +s 1s )))
2726rexbidv 3181 . . 3 (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))
2825, 27orbi12d 917 . 2 (𝑚 = 𝑁 → ((∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s )) ↔ (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s ))))
29 0n0s 28343 . . . 4 0s ∈ ℕ0s
30 2sno 28412 . . . . . 6 2s No
31 muls01 28147 . . . . . 6 (2s No → (2s ·s 0s ) = 0s )
3230, 31ax-mp 5 . . . . 5 (2s ·s 0s ) = 0s
3332eqcomi 2743 . . . 4 0s = (2s ·s 0s )
34 oveq2 7453 . . . . 5 (𝑥 = 0s → (2s ·s 𝑥) = (2s ·s 0s ))
3534rspceeqv 3653 . . . 4 (( 0s ∈ ℕ0s ∧ 0s = (2s ·s 0s )) → ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥))
3629, 33, 35mp2an 691 . . 3 𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥)
3736orci 864 . 2 (∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 0s = ((2s ·s 𝑥) +s 1s ))
38 eqid 2734 . . . . . . . 8 ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑥) +s 1s )
39 oveq2 7453 . . . . . . . . . 10 (𝑦 = 𝑥 → (2s ·s 𝑦) = (2s ·s 𝑥))
4039oveq1d 7460 . . . . . . . . 9 (𝑦 = 𝑥 → ((2s ·s 𝑦) +s 1s ) = ((2s ·s 𝑥) +s 1s ))
4140rspceeqv 3653 . . . . . . . 8 ((𝑥 ∈ ℕ0s ∧ ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑥) +s 1s )) → ∃𝑦 ∈ ℕ0s ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
4238, 41mpan2 690 . . . . . . 7 (𝑥 ∈ ℕ0s → ∃𝑦 ∈ ℕ0s ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
43 oveq1 7452 . . . . . . . . 9 (𝑛 = (2s ·s 𝑥) → (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ))
4443eqeq1d 2736 . . . . . . . 8 (𝑛 = (2s ·s 𝑥) → ((𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ) ↔ ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s )))
4544rexbidv 3181 . . . . . . 7 (𝑛 = (2s ·s 𝑥) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ) ↔ ∃𝑦 ∈ ℕ0s ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s )))
4642, 45syl5ibrcom 247 . . . . . 6 (𝑥 ∈ ℕ0s → (𝑛 = (2s ·s 𝑥) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
4746rexlimiv 3150 . . . . 5 (∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))
48 peano2n0s 28344 . . . . . . . 8 (𝑥 ∈ ℕ0s → (𝑥 +s 1s ) ∈ ℕ0s)
49 1p1e2s 28409 . . . . . . . . . . 11 ( 1s +s 1s ) = 2s
50 mulsrid 28148 . . . . . . . . . . . 12 (2s No → (2s ·s 1s ) = 2s)
5130, 50ax-mp 5 . . . . . . . . . . 11 (2s ·s 1s ) = 2s
5249, 51eqtr4i 2765 . . . . . . . . . 10 ( 1s +s 1s ) = (2s ·s 1s )
5352oveq2i 7456 . . . . . . . . 9 ((2s ·s 𝑥) +s ( 1s +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s ))
5430a1i 11 . . . . . . . . . . 11 (𝑥 ∈ ℕ0s → 2s No )
55 n0sno 28337 . . . . . . . . . . 11 (𝑥 ∈ ℕ0s𝑥 No )
5654, 55mulscld 28170 . . . . . . . . . 10 (𝑥 ∈ ℕ0s → (2s ·s 𝑥) ∈ No )
57 1sno 27881 . . . . . . . . . . 11 1s No
5857a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ0s → 1s No )
5956, 58, 58addsassd 28048 . . . . . . . . 9 (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = ((2s ·s 𝑥) +s ( 1s +s 1s )))
6054, 55, 58addsdid 28191 . . . . . . . . 9 (𝑥 ∈ ℕ0s → (2s ·s (𝑥 +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s )))
6153, 59, 603eqtr4a 2800 . . . . . . . 8 (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s (𝑥 +s 1s )))
62 oveq2 7453 . . . . . . . . 9 (𝑦 = (𝑥 +s 1s ) → (2s ·s 𝑦) = (2s ·s (𝑥 +s 1s )))
6362rspceeqv 3653 . . . . . . . 8 (((𝑥 +s 1s ) ∈ ℕ0s ∧ (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s (𝑥 +s 1s ))) → ∃𝑦 ∈ ℕ0s (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦))
6448, 61, 63syl2anc 583 . . . . . . 7 (𝑥 ∈ ℕ0s → ∃𝑦 ∈ ℕ0s (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦))
65 oveq1 7452 . . . . . . . . 9 (𝑛 = ((2s ·s 𝑥) +s 1s ) → (𝑛 +s 1s ) = (((2s ·s 𝑥) +s 1s ) +s 1s ))
6665eqeq1d 2736 . . . . . . . 8 (𝑛 = ((2s ·s 𝑥) +s 1s ) → ((𝑛 +s 1s ) = (2s ·s 𝑦) ↔ (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦)))
6766rexbidv 3181 . . . . . . 7 (𝑛 = ((2s ·s 𝑥) +s 1s ) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ↔ ∃𝑦 ∈ ℕ0s (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦)))
6864, 67syl5ibrcom 247 . . . . . 6 (𝑥 ∈ ℕ0s → (𝑛 = ((2s ·s 𝑥) +s 1s ) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
6968rexlimiv 3150 . . . . 5 (∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s ) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦))
7047, 69orim12i 907 . . . 4 ((∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦)))
7170orcomd 870 . . 3 ((∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
7271a1i 11 . 2 (𝑛 ∈ ℕ0s → ((∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑛 = ((2s ·s 𝑥) +s 1s )) → (∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑦) ∨ ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))))
735, 10, 23, 28, 37, 72n0sind 28346 1 (𝑁 ∈ ℕ0s → (∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥) ∨ ∃𝑥 ∈ ℕ0s 𝑁 = ((2s ·s 𝑥) +s 1s )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846   = wceq 1537  wcel 2103  wrex 3072  (class class class)co 7445   No csur 27693   0s c0s 27876   1s c1s 27877   +s cadds 28001   ·s cmuls 28141  0scnn0s 28327  2sc2s 28403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-2o 8519  df-nadd 8718  df-no 27696  df-slt 27697  df-bday 27698  df-sle 27799  df-sslt 27835  df-scut 27837  df-0s 27878  df-1s 27879  df-made 27895  df-old 27896  df-left 27898  df-right 27899  df-norec 27980  df-norec2 27991  df-adds 28002  df-negs 28062  df-subs 28063  df-muls 28142  df-n0s 28329  df-nns 28330  df-2s 28404
This theorem is referenced by:  zseo  28415
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