Theorem bdayfinbnd 28479

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 6836	. . . . . 6 ⊢ (𝑧 = 𝑍 → ( bday ‘𝑧) = ( bday ‘𝑍))
4	3	sseq1d 3954	. . . . 5 ⊢ (𝑧 = 𝑍 → (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ↔ ( bday ‘𝑍) ⊆ ( bday ‘𝑁)))
5		breq2 5090	. . . . 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 28478	. . . 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 3566	. 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 3890   class class class wbr 5086  ‘cfv 6494  (class class class)co 7362   No csur 27621   <s clts 27622   bday cbday 27623   ≤s cles 27726   0_s c0s 27815   +_s cadds 27969   /_su cdivs 28197  ℕ_0scn0s 28322  2_sc2s 28420  ↑_scexps 28422

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-dc 10363  ax-ac2 10380

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-nadd 8597  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-fin 8892  df-card 9858  df-acn 9861  df-ac 10033  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-muls 28117  df-divs 28198  df-ons 28262  df-seqs 28294  df-n0s 28324  df-nns 28325  df-zs 28389  df-2s 28421  df-exps 28423

This theorem is referenced by:  bdayfinlem  28496