Theorem bdayfinlem 28872

Description: Lemma for bdayfin 28873. 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 28756	. . . . . . 7 ⊢ (bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
2		f1ocnvdm 7293	. . . . . . 7 ⊢ (((bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ (bday‘𝐴) ∈ ω) → (◡(bday ↾ ℕ0s)‘(bday‘𝐴)) ∈ ℕ0s)
3	1, 2	mpan 703	. . . . . 6 ⊢ ((bday‘𝐴) ∈ ω → (◡(bday ↾ ℕ0s)‘(bday‘𝐴)) ∈ ℕ0s)
4	3	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → (◡(bday ↾ ℕ0s)‘(bday‘𝐴)) ∈ ℕ0s)
5	4	n0zsd 28776	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → (◡(bday ↾ ℕ0s)‘(bday‘𝐴)) ∈ ℤs)
6		zz12s 28861	. . . 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 2848	. . 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 28775	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
11		zz12s 28861	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ ℤs[1/2])
12	10, 11	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs[1/2])
13	12	adantr 486	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → 𝑥 ∈ ℤs[1/2])
14		n0zs 28775	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℤs)
15		elz12si 28859	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
16	14, 15	sylan 592	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
17	16	adantll 727	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
18		z12addscl 28863	. . . . . . 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 698	. . . . . 6 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2])
20		eleq1 2848	. . . . . . 7 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) → (𝐴 ∈ ℤs[1/2] ↔ (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2]))
21	20	3ad2ant1 1151	. . . . . 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 3163	. . . 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 3204	. . 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 1154	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → 𝐴 ∈ No)
27	4	fvresd 6905	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → ((bday ↾ ℕ0s)‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))) = (bday‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))))
28		f1ocnvfv2 7285	. . . . . . 7 ⊢ (((bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ (bday‘𝐴) ∈ ω) → ((bday ↾ ℕ0s)‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))) = (bday‘𝐴))
29	1, 28	mpan 703	. . . . . 6 ⊢ ((bday‘𝐴) ∈ ω → ((bday ↾ ℕ0s)‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))) = (bday‘𝐴))
30	29	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → ((bday ↾ ℕ0s)‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))) = (bday‘𝐴))
31	27, 30	eqtr3d 2797	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → (bday‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))) = (bday‘𝐴))
32	31	eqimsscd 3965	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → (bday‘𝐴) ⊆ (bday‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))))
33		simp2 1155	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → 0s ≤s 𝐴)
34	4, 26, 32, 33	bdayfinbnd 28855	. 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 874	1 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  ^◡ccnv 5649   ↾ cres 5652  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420  ωcom 7877  Nocsur 27997   <s clts 27998  bdaycbday 27999   ≤s cles 28101   0_s c0s 28191   +_s cadds 28345   /_su cdivs 28573  ℕ_0scn0s 28698  ℤ_sczs 28764  2_sc2s 28796  ↑_scexps 28798  ℤ_s[1/2]cz12s 28800

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  df-z12s 28801

This theorem is used by:  bdayfin  28873