Theorem bdayfinlem 28463

Description: Lemma for bdayfin 28464. 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 28347	. . . . . . 7 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
2		f1ocnvdm 7231	. . . . . . 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 28367	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs)
6		zzs12 28452	. . . 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 2823	. . 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 28366	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
11		zzs12 28452	. . . . . . . . 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 28366	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℤs)
15		elzs12i 28450	. . . . . . . . 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		zs12addscl 28454	. . . . . . 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 2823	. . . . . . 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 3136	. . . 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 3177	. . 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 6853	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))))
28		f1ocnvfv2 7223	. . . . . . 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 2772	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘𝐴))
32	31	eqimsscd 3990	. . 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 28446	. 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 3059   class class class wbr 5097  ^◡ccnv 5622   ↾ cres 5625  –1-1-onto→wf1o 6490  ‘cfv 6491  (class class class)co 7358  ωcom 7808   No csur 27609   <s cslt 27610   bday cbday 27611   ≤s csle 27714   0_s c0s 27801   +_s cadds 27939   /_su cdivs 28167  ℕ_0scnn0s 28291  ℤ_sczs 28355  2_sc2s 28387  ↑_scexps 28389  ℤ_s[1/2]czs12 28391

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-dc 10358  ax-ac2 10375

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-er 8635  df-map 8767  df-en 8886  df-dom 8887  df-fin 8889  df-card 9853  df-acn 9856  df-ac 10028  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-muls 28087  df-divs 28168  df-ons 28231  df-seqs 28263  df-n0s 28293  df-nns 28294  df-zs 28356  df-2s 28388  df-exps 28390  df-zs12 28392

This theorem is referenced by:  bdayfin  28464