Theorem bdayfinbnd 28482

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 6844	. . . . . 6 ⊢ (𝑧 = 𝑍 → ( bday ‘𝑧) = ( bday ‘𝑍))
4	3	sseq1d 3967	. . . . 5 ⊢ (𝑧 = 𝑍 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑍) ⊆ ( bday ‘𝑁)))
5		breq2 5104	. . . . 5 ⊢ (𝑧 = 𝑍 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑍))
6	4, 5	anbi12d 633	. . . 4 ⊢ (𝑧 = 𝑍 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑍) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑍)))
7		eqeq1 2741	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑧 = 𝑁 ↔ 𝑍 = 𝑁))
8		eqeq1 2741	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑍 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
9	8	3anbi1d 1443	. . . . . . 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 3162	. . . . . 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 3203	. . . . 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 919	. . . 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 344	. . 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 28481	. . . 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 3579	. 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 700	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 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903   class class class wbr 5100  ‘cfv 6502  (class class class)co 7370   No csur 27624   <s clts 27625   bday cbday 27626   ≤s cles 27729   0_s c0s 27818   +_s cadds 27972   /_su cdivs 28200  ℕ_0scn0s 28325  2_sc2s 28423  ↑_scexps 28425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-dc 10370  ax-ac2 10387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-2o 8410  df-oadd 8413  df-nadd 8606  df-er 8647  df-map 8779  df-en 8898  df-dom 8899  df-fin 8901  df-card 9865  df-acn 9868  df-ac 10040  df-no 27627  df-lts 27628  df-bday 27629  df-les 27730  df-slts 27771  df-cuts 27773  df-0s 27820  df-1s 27821  df-made 27840  df-old 27841  df-left 27843  df-right 27844  df-norec 27951  df-norec2 27962  df-adds 27973  df-negs 28034  df-subs 28035  df-muls 28120  df-divs 28201  df-ons 28265  df-seqs 28297  df-n0s 28327  df-nns 28328  df-zs 28392  df-2s 28424  df-exps 28426

This theorem is referenced by:  bdayfinlem  28499