Theorem bdayfinlem 28500

Description: Lemma for bdayfin 28501. 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 28384	. . . . . . 7 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
2		f1ocnvdm 7233	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
3	1, 2	mpan 697	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ ω → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
4	3	3ad2ant3 1142	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℕ0s)
5	4	n0zsd 28404	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs)
6		zz12s 28489	. . . 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 2829	. . 3 ⊢ (𝐴 = (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) → (𝐴 ∈ ℤs[1/2] ↔ (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs[1/2]))
9	7, 8	syl5ibrcom 249	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (𝐴 = (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) → 𝐴 ∈ ℤs[1/2]))
10		n0zs 28403	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
11		zz12s 28489	. . . . . . . . 9 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ ℤs[1/2])
12	10, 11	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs[1/2])
13	12	adantr 482	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → 𝑥 ∈ ℤs[1/2])
14		n0zs 28403	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℤs)
15		elz12si 28487	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℤs ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
16	14, 15	sylan 587	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
17	16	adantll 721	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑦 /su (2s↑s𝑧)) ∈ ℤs[1/2])
18		z12addscl 28491	. . . . . . 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 692	. . . . . 6 ⊢ (((𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2])
20		eleq1 2829	. . . . . . 7 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑧))) → (𝐴 ∈ ℤs[1/2] ↔ (𝑥 +s (𝑦 /su (2s↑s𝑧))) ∈ ℤs[1/2]))
21	20	3ad2ant1 1140	. . . . . 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 249	. . . . 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 3142	. . . 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 3183	. . 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 1143	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ No )
27	4	fvresd 6851	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))))
28		f1ocnvfv2 7225	. . . . . . 7 ⊢ ((( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
29	1, 28	mpan 697	. . . . . 6 ⊢ (( bday ‘𝐴) ∈ ω → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
30	29	3ad2ant3 1142	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
31	27, 30	eqtr3d 2778	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
32	31	eqimsscd 3974	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → ( bday ‘𝐴) ⊆ ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))))
33		simp2 1144	. . 3 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 0s ≤s 𝐴)
34	4, 26, 32, 33	bdayfinbnd 28483	. 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 867	1 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → 𝐴 ∈ ℤs[1/2])

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∃wrex 3065   class class class wbr 5075  ^◡ccnv 5620   ↾ cres 5623  –1-1-onto→wf1o 6488  ‘cfv 6489  (class class class)co 7360  ωcom 7810   No csur 27625   <s clts 27626   bday cbday 27627   ≤s cles 27730   0_s c0s 27819   +_s cadds 27973   /_su cdivs 28201  ℕ_0scn0s 28326  ℤ_sczs 28392  2_sc2s 28424  ↑_scexps 28426  ℤ_s[1/2]cz12s 28428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-dc 10363  ax-ac2 10380

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-ot 4567  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-isom 6498  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-fin 8891  df-card 9858  df-acn 9861  df-ac 10033  df-no 27628  df-lts 27629  df-bday 27630  df-les 27731  df-slts 27772  df-cuts 27774  df-0s 27821  df-1s 27822  df-made 27841  df-old 27842  df-left 27844  df-right 27845  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28035  df-subs 28036  df-muls 28121  df-divs 28202  df-ons 28266  df-seqs 28298  df-n0s 28328  df-nns 28329  df-zs 28393  df-2s 28425  df-exps 28427  df-z12s 28429

This theorem is referenced by:  bdayfin  28501