Theorem bdayfinlem 28499

Description: Lemma for bdayfin 28500. 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 28383	. . . . . . 7 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
2		f1ocnvdm 7243	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
3	1, 2	mpan 691	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ ω → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
4	3	3ad2ant3 1136	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
5	4	n0zsd 28403	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs)
6		zz12s 28488	. . . 4 ⊢ ((◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs[1/2])
7	5, 6	syl 17	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs[1/2])
8		eleq1 2825	. . 3 ⊢ (𝐴 = (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) → (𝐴 ∈ ℤs[1/2] ↔ (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs[1/2]))
9	7, 8	syl5ibrcom 247	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (𝐴 = (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) → 𝐴 ∈ ℤs[1/2]))
10		n0zs 28402	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
11		zz12s 28488	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ ℤs[1/2])
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs[1/2])
13	12	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → 𝑥 ∈ ℤs[1/2])
14		n0zs 28402	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℤs)
15		elz12si 28486	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
16	14, 15	sylan 581	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
17	16	adantll 715	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
18		z12addscl 28490	. . . . . . 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 686	. . . . . 6 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2])
20		eleq1 2825	. . . . . . 7 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) → (𝐴 ∈ ℤs[1/2] ↔ (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2]))
21	20	3ad2ant1 1134	. . . . . 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 247	. . . . 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 3139	. . . 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 3180	. . 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 1137	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ No )
27	4	fvresd 6864	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))))
28		f1ocnvfv2 7235	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
29	1, 28	mpan 691	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ ω → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
30	29	3ad2ant3 1136	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
31	27, 30	eqtr3d 2774	. . . 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 1138	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 0s ≤s 𝐴)
34	4, 26, 32, 33	bdayfinbnd 28482	. 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 861	1 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062   class class class wbr 5100  ^◡ccnv 5633   ↾ cres 5636  –1-1-onto→wf1o 6501  ‘cfv 6502  (class class class)co 7370  ωcom 7820   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  ℤ_sczs 28391  2_sc2s 28423  ↑_scexps 28425  ℤ_s[1/2]cz12s 28427

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  df-z12s 28428

This theorem is referenced by:  bdayfin  28500