Theorem bdayfinlem 28690

Description: Lemma for bdayfin 28691. Handle the non-negative case. (Contributed by Scott Fenton, 26-Feb-2026.)

Assertion

Ref	Expression
bdayfinlem	⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])

Proof of Theorem bdayfinlem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayn0sf1o 28574	. . . . . . 7 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
2		f1ocnvdm 7283	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
3	1, 2	mpan 702	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ ω → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
4	3	3ad2ant3 1152	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
5	4	n0zsd 28594	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs)
6		zz12s 28679	. . . 4 ⊢ ((◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs[1/2])
7	5, 6	syl 18	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs[1/2])
8		eleq1 2850	. . 3 ⊢ (𝐴 = (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) → (𝐴 ∈ ℤs[1/2] ↔ (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs[1/2]))
9	7, 8	syl5ibrcom 250	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (𝐴 = (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) → 𝐴 ∈ ℤs[1/2]))
10		n0zs 28593	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
11		zz12s 28679	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ ℤs[1/2])
12	10, 11	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs[1/2])
13	12	adantr 485	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → 𝑥 ∈ ℤs[1/2])
14		n0zs 28593	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℤs)
15		elz12si 28677	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
16	14, 15	sylan 591	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
17	16	adantll 726	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
18		z12addscl 28681	. . . . . . 7 ⊢ ((𝑥 ∈ ℤs[1/2] ∧ (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2]) → (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2])
19	13, 17, 18	syl2an2r 697	. . . . . 6 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2])
20		eleq1 2850	. . . . . . 7 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) → (𝐴 ∈ ℤs[1/2] ↔ (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2]))
21	20	3ad2ant1 1150	. . . . . 6 ⊢ ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∧ 𝑦 <s (2s↑s𝑧) ∧ (𝑥 +s 𝑧) <s (◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) → (𝐴 ∈ ℤs[1/2] ↔ (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2]))
22	19, 21	syl5ibrcom 250	. . . . 5 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∧ 𝑦 <s (2s↑s𝑧) ∧ (𝑥 +s 𝑧) <s (◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) → 𝐴 ∈ ℤs[1/2]))
23	22	rexlimdva 3165	. . . 4 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (∃𝑧 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∧ 𝑦 <s (2s↑s𝑧) ∧ (𝑥 +s 𝑧) <s (◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) → 𝐴 ∈ ℤs[1/2]))
24	23	rexlimivv 3206	. . 3 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∧ 𝑦 <s (2s↑s𝑧) ∧ (𝑥 +s 𝑧) <s (◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) → 𝐴 ∈ ℤs[1/2])
25	24	a1i 11	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∧ 𝑦 <s (2s↑s𝑧) ∧ (𝑥 +s 𝑧) <s (◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) → 𝐴 ∈ ℤs[1/2]))
26		simp1 1153	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ No )
27	4	fvresd 6901	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))))
28		f1ocnvfv2 7275	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
29	1, 28	mpan 702	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ ω → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
30	29	3ad2ant3 1152	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
31	27, 30	eqtr3d 2799	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
32	31	eqimsscd 3993	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → ( bday ‘𝐴) ⊆ ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))))
33		simp2 1154	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 0s ≤s 𝐴)
34	4, 26, 32, 33	bdayfinbnd 28673	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (𝐴 = (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∧ 𝑦 <s (2s↑s𝑧) ∧ (𝑥 +s 𝑧) <s (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)))))
35	9, 25, 34	mpjaod 873	1 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   class class class wbr 5108  ^◡ccnv 5659   ↾ cres 5662  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7412  ωcom 7860   No csur 27815   <s clts 27816   bday cbday 27817   ≤s cles 27919   0_s c0s 28009   +_s cadds 28163   /_su cdivs 28391  ℕ_0scn0s 28516  ℤ_sczs 28582  2_sc2s 28614  ↑_scexps 28616  ℤ_s[1/2]cz12s 28618

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-dc 10436  ax-ac2 10453

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-oadd 8455  df-nadd 8650  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-fin 8945  df-card 9932  df-acn 9935  df-ac 10107  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311  df-divs 28392  df-ons 28456  df-seqs 28488  df-n0s 28518  df-nns 28519  df-zs 28583  df-2s 28615  df-exps 28617  df-z12s 28619

This theorem is used by:  bdayfin  28691