Theorem bdayfinbnd 28483

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 6831	. . . . . 6 ⊢ (𝑧 = 𝑍 → ( bday ‘𝑧) = ( bday ‘𝑍))
4	3	sseq1d 3948	. . . . 5 ⊢ (𝑧 = 𝑍 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑍) ⊆ ( bday ‘𝑁)))
5		breq2 5079	. . . . 5 ⊢ (𝑧 = 𝑍 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑍))
6	4, 5	anbi12d 639	. . . 4 ⊢ (𝑧 = 𝑍 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑍) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑍)))
7		eqeq1 2745	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑁 ↔ 𝑍 = 𝑁))
8		eqeq1 2745	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
9	8	3anbi1d 1449	. . . . . . 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 3165	. . . . . 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 3206	. . . . 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 925	. . . 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 346	. . 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 28482	. . . 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 17	. . 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 3563	. 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 706	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)))

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065   ⊆ wss 3885   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360   No csur 27625   <s clts 27626   bday cbday 27627   ≤s cles 27730   0_s c0s 27819   +_s cadds 27973   /_su cdivs 28201  ℕ_0scn0s 28326  2_sc2s 28424  ↑_scexps 28426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-dc 10363  ax-ac2 10380

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-fin 8891  df-card 9858  df-acn 9861  df-ac 10033  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-0s 27821  df-1s 27822  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28035  df-subs 28036  df-muls 28121  df-divs 28202  df-ons 28266  df-seqs 28298  df-n0s 28328  df-nns 28329  df-zs 28393  df-2s 28425  df-exps 28427

This theorem is referenced by:  bdayfinlem  28500