Theorem bdayfinbnd 28882

Description: Given a non-negative integer and a non-negative surreal of lesser or equal birthday, show that the surreal can be expressed as a dyadic fraction with an upper bound on the integer and exponent. This proof follows the proof from Mizar at https://mizar.uwb.edu.pl/version/current/html/surrealn.html. (Contributed by Scott Fenton, 26-Feb-2026.)

Hypotheses

Ref	Expression
bdayfinbnd.1	⊢ (𝜑 → 𝑁 ∈ ℕ0s)
bdayfinbnd.2	⊢ (𝜑 → 𝑍 ∈ No)
bdayfinbnd.3	⊢ (𝜑 → (bday‘𝑍) ⊆ (bday‘𝑁))
bdayfinbnd.4	⊢ (𝜑 → 0s ≤s 𝑍)

Assertion

Ref	Expression
bdayfinbnd	⊢ (𝜑 → (𝑍 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))

Distinct variable groups:   𝑥,𝑁,𝑦,𝑝   𝑥,𝑍,𝑦,𝑝

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑝)

Proof of Theorem bdayfinbnd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayfinbnd.3	. 2 ⊢ (𝜑 → (bday‘𝑍) ⊆ (bday‘𝑁))
2		bdayfinbnd.4	. 2 ⊢ (𝜑 → 0s ≤s 𝑍)
3		fveq2 6905	. . . . . 6 ⊢ (𝑧 = 𝑍 → (bday‘𝑧) = (bday‘𝑍))
4	3	sseq1d 3944	. . . . 5 ⊢ (𝑧 = 𝑍 → ((bday‘𝑧) ⊆ (bday‘𝑁) ↔ (bday‘𝑍) ⊆ (bday‘𝑁)))
5		breq2 5110	. . . . 5 ⊢ (𝑧 = 𝑍 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑍))
6	4, 5	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝑍 → (((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) ↔ ((bday‘𝑍) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑍)))
7		eqeq1 2767	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑁 ↔ 𝑍 = 𝑁))
8		eqeq1 2767	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
9	8	3anbi1d 1469	. . . . . . 7 ⊢ (𝑧 = 𝑍 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
10	9	rexbidv 3190	. . . . . 6 ⊢ (𝑧 = 𝑍 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
11	10	2rexbidv 3231	. . . . 5 ⊢ (𝑧 = 𝑍 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
12	7, 11	orbi12d 932	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)) ↔ (𝑍 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
13	6, 12	imbi12d 347	. . 3 ⊢ (𝑧 = 𝑍 → ((((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))) ↔ (((bday‘𝑍) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑍) → (𝑍 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))))
14		bdayfinbnd.1	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
15		bdayfinbndlem2 28881	. . . 4 ⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
17		bdayfinbnd.2	. . 3 ⊢ (𝜑 → 𝑍 ∈ No)
18	13, 16, 17	rspcdva 3581	. 2 ⊢ (𝜑 → (((bday‘𝑍) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑍) → (𝑍 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
19	1, 2, 18	mp2and 712	1 ⊢ (𝜑 → (𝑍 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5106  ‘cfv 6555  (class class class)co 7440  Nocsur 28024   <s clts 28025  bdaycbday 28026   ≤s cles 28128   0_s c0s 28218   +_s cadds 28372   /_su cdivs 28600  ℕ_0scn0s 28725  2_sc2s 28823  ↑_scexps 28825

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-dc 10548

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-nadd 8700  df-en 8999  df-dom 9000  df-fin 9002  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-1s 28221  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-muls 28520  df-divs 28601  df-ons 28665  df-seqs 28697  df-n0s 28727  df-nns 28728  df-zs 28792  df-2s 28824  df-exps 28826

This theorem is used by:  bdayfinlem  28899