Theorem bdayfinbnd 28855

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 6885	. . . . . 6 ⊢ (𝑧 = 𝑍 → (bday‘𝑧) = (bday‘𝑍))
4	3	sseq1d 3939	. . . . 5 ⊢ (𝑧 = 𝑍 → ((bday‘𝑧) ⊆ (bday‘𝑁) ↔ (bday‘𝑍) ⊆ (bday‘𝑁)))
5		breq2 5106	. . . . 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 2764	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑁 ↔ 𝑍 = 𝑁))
8		eqeq1 2764	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
9	8	3anbi1d 1468	. . . . . . 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 3186	. . . . . 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 3227	. . . . 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 28854	. . . 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 3577	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   <s clts 27998  bdaycbday 27999   ≤s cles 28101   0_s c0s 28191   +_s cadds 28345   /_su cdivs 28573  ℕ_0scn0s 28698  2_sc2s 28796  ↑_scexps 28798

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-dc 10524

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-nadd 8675  df-en 8974  df-dom 8975  df-fin 8977  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493  df-divs 28574  df-ons 28638  df-seqs 28670  df-n0s 28700  df-nns 28701  df-zs 28765  df-2s 28797  df-exps 28799

This theorem is used by:  bdayfinlem  28872