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Theorem n0seo 28423
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 2744 . . . 4 (𝑚 = 0s → (𝑚 = (2s ·s 𝑥) ↔ 0s = (2s ·s 𝑥)))
21rexbidv 3185 . . 3 (𝑚 = 0s → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 0s = (2s ·s 𝑥)))
3 eqeq1 2744 . . . 4 (𝑚 = 0s → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 0s = ((2s ·s 𝑥) +s 1s )))
43rexbidv 3185 . . 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 2744 . . . 4 (𝑚 = 𝑛 → (𝑚 = (2s ·s 𝑥) ↔ 𝑛 = (2s ·s 𝑥)))
76rexbidv 3185 . . 3 (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥)))
8 eqeq1 2744 . . . 4 (𝑚 = 𝑛 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑛 = ((2s ·s 𝑥) +s 1s )))
98rexbidv 3185 . . 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 2744 . . . . 5 (𝑚 = (𝑛 +s 1s ) → (𝑚 = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑥)))
1211rexbidv 3185 . . . 4 (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = (2s ·s 𝑥)))
13 oveq2 7456 . . . . . 6 (𝑥 = 𝑦 → (2s ·s 𝑥) = (2s ·s 𝑦))
1413eqeq2d 2751 . . . . 5 (𝑥 = 𝑦 → ((𝑛 +s 1s ) = (2s ·s 𝑥) ↔ (𝑛 +s 1s ) = (2s ·s 𝑦)))
1514cbvrexvw 3244 . . . 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 2744 . . . . 5 (𝑚 = (𝑛 +s 1s ) → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s )))
1817rexbidv 3185 . . . 4 (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s 𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ ∃𝑥 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s )))
1913oveq1d 7463 . . . . . 6 (𝑥 = 𝑦 → ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s ))
2019eqeq2d 2751 . . . . 5 (𝑥 = 𝑦 → ((𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ) ↔ (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s )))
2120cbvrexvw 3244 . . . 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 2744 . . . 4 (𝑚 = 𝑁 → (𝑚 = (2s ·s 𝑥) ↔ 𝑁 = (2s ·s 𝑥)))
2524rexbidv 3185 . . 3 (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s 𝑚 = (2s ·s 𝑥) ↔ ∃𝑥 ∈ ℕ0s 𝑁 = (2s ·s 𝑥)))
26 eqeq1 2744 . . . 4 (𝑚 = 𝑁 → (𝑚 = ((2s ·s 𝑥) +s 1s ) ↔ 𝑁 = ((2s ·s 𝑥) +s 1s )))
2726rexbidv 3185 . . 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 28352 . . . 4 0s ∈ ℕ0s
30 2sno 28421 . . . . . 6 2s No
31 muls01 28156 . . . . . 6 (2s No → (2s ·s 0s ) = 0s )
3230, 31ax-mp 5 . . . . 5 (2s ·s 0s ) = 0s
3332eqcomi 2749 . . . 4 0s = (2s ·s 0s )
34 oveq2 7456 . . . . 5 (𝑥 = 0s → (2s ·s 𝑥) = (2s ·s 0s ))
3534rspceeqv 3658 . . . 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 2740 . . . . . . . 8 ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑥) +s 1s )
39 oveq2 7456 . . . . . . . . . 10 (𝑦 = 𝑥 → (2s ·s 𝑦) = (2s ·s 𝑥))
4039oveq1d 7463 . . . . . . . . 9 (𝑦 = 𝑥 → ((2s ·s 𝑦) +s 1s ) = ((2s ·s 𝑥) +s 1s ))
4140rspceeqv 3658 . . . . . . . 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 7455 . . . . . . . . 9 (𝑛 = (2s ·s 𝑥) → (𝑛 +s 1s ) = ((2s ·s 𝑥) +s 1s ))
4443eqeq1d 2742 . . . . . . . 8 (𝑛 = (2s ·s 𝑥) → ((𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ) ↔ ((2s ·s 𝑥) +s 1s ) = ((2s ·s 𝑦) +s 1s )))
4544rexbidv 3185 . . . . . . 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 3154 . . . . 5 (∃𝑥 ∈ ℕ0s 𝑛 = (2s ·s 𝑥) → ∃𝑦 ∈ ℕ0s (𝑛 +s 1s ) = ((2s ·s 𝑦) +s 1s ))
48 peano2n0s 28353 . . . . . . . 8 (𝑥 ∈ ℕ0s → (𝑥 +s 1s ) ∈ ℕ0s)
49 1p1e2s 28418 . . . . . . . . . . 11 ( 1s +s 1s ) = 2s
50 mulsrid 28157 . . . . . . . . . . . 12 (2s No → (2s ·s 1s ) = 2s)
5130, 50ax-mp 5 . . . . . . . . . . 11 (2s ·s 1s ) = 2s
5249, 51eqtr4i 2771 . . . . . . . . . 10 ( 1s +s 1s ) = (2s ·s 1s )
5352oveq2i 7459 . . . . . . . . 9 ((2s ·s 𝑥) +s ( 1s +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s ))
5430a1i 11 . . . . . . . . . . 11 (𝑥 ∈ ℕ0s → 2s No )
55 n0sno 28346 . . . . . . . . . . 11 (𝑥 ∈ ℕ0s𝑥 No )
5654, 55mulscld 28179 . . . . . . . . . 10 (𝑥 ∈ ℕ0s → (2s ·s 𝑥) ∈ No )
57 1sno 27890 . . . . . . . . . . 11 1s No
5857a1i 11 . . . . . . . . . 10 (𝑥 ∈ ℕ0s → 1s No )
5956, 58, 58addsassd 28057 . . . . . . . . 9 (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = ((2s ·s 𝑥) +s ( 1s +s 1s )))
6054, 55, 58addsdid 28200 . . . . . . . . 9 (𝑥 ∈ ℕ0s → (2s ·s (𝑥 +s 1s )) = ((2s ·s 𝑥) +s (2s ·s 1s )))
6153, 59, 603eqtr4a 2806 . . . . . . . 8 (𝑥 ∈ ℕ0s → (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s (𝑥 +s 1s )))
62 oveq2 7456 . . . . . . . . 9 (𝑦 = (𝑥 +s 1s ) → (2s ·s 𝑦) = (2s ·s (𝑥 +s 1s )))
6362rspceeqv 3658 . . . . . . . 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 7455 . . . . . . . . 9 (𝑛 = ((2s ·s 𝑥) +s 1s ) → (𝑛 +s 1s ) = (((2s ·s 𝑥) +s 1s ) +s 1s ))
6665eqeq1d 2742 . . . . . . . 8 (𝑛 = ((2s ·s 𝑥) +s 1s ) → ((𝑛 +s 1s ) = (2s ·s 𝑦) ↔ (((2s ·s 𝑥) +s 1s ) +s 1s ) = (2s ·s 𝑦)))
6766rexbidv 3185 . . . . . . 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 3154 . . . . 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 28355 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 2108  wrex 3076  (class class class)co 7448   No csur 27702   0s c0s 27885   1s c1s 27886   +s cadds 28010   ·s cmuls 28150  0scnn0s 28336  2sc2s 28412
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 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
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 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  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 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-muls 28151  df-n0s 28338  df-nns 28339  df-2s 28413
This theorem is referenced by:  zseo  28424
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