Theorem bdayfinlem 28899

Description: Lemma for bdayfin 28900. 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 28783	. . . . . . 7 ⊢ (bday ↾ ℕ0s):ℕ0s⟶1-1ontoω
2		f1ocnvdm 7313	. . . . . . 7 ⊢ (((bday ↾ ℕ0s):ℕ0s⟶1-1ontoω ∧ (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 28803	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → (◡(bday ↾ ℕ0s)‘(bday‘𝐴)) ∈ ℤs)
6		zz12s 28888	. . . 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 2852	. . 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 28802	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
11		zz12s 28888	. . . . . . . . 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 28802	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℤs)
15		elz12si 28886	. . . . . . . . 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 28890	. . . . . . 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 2852	. . . . . . 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 3167	. . . 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 3208	. . 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 6925	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → ((bday ↾ ℕ0s)‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))) = (bday‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))))
28		f1ocnvfv2 7305	. . . . . . 7 ⊢ (((bday ↾ ℕ0s):ℕ0s⟶1-1ontoω ∧ (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 2800	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ (bday‘𝐴) ∈ ω) → (bday‘(◡(bday ↾ ℕ0s)‘(bday‘𝐴))) = (bday‘𝐴))
32	31	eqimsscd 3970	. . 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 28882	. 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 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5106  ^◡ccnv 5654   ↾ cres 5657  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  ωcom 7898  Nocsur 28024   <s clts 28025  bdaycbday 28026   ≤s cles 28128   0_s c0s 28218   +_s cadds 28372   /_su cdivs 28600  ℕ_0scn0s 28725  ℤ_sczs 28791  2_sc2s 28823  ↑_scexps 28825  ℤ_s[1/2]cz12s 28827

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-dc 10548

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-nadd 8700  df-en 8999  df-dom 9000  df-fin 9002  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-1s 28221  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-muls 28520  df-divs 28601  df-ons 28665  df-seqs 28697  df-n0s 28727  df-nns 28728  df-zs 28792  df-2s 28824  df-exps 28826  df-z12s 28828

This theorem is used by:  bdayfin  28900