Theorem bdayfinlem 28496

Description: Lemma for bdayfin 28497. 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 28380	. . . . . . 7 ⊢ ( bday ↾ ℕ0s):ℕ0s–1-1-onto→ω
2		f1ocnvdm 7235	. . . . . . 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 28400	. . . 4 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (◡( bday ↾ ℕ0s)‘( bday ‘𝐴)) ∈ ℤs)
6		zz12s 28485	. . . 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 28399	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0s → 𝑥 ∈ ℤs)
11		zz12s 28485	. . . . . . . . 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 28399	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℤs)
15		elz12si 28483	. . . . . . . . 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 28487	. . . . . . 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 6856	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴 ∧ ( bday ‘𝐴) ∈ ω) → (( bday ↾ ℕ0s)‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))) = ( bday ‘(◡( bday ↾ ℕ0s)‘( bday ‘𝐴))))
28		f1ocnvfv2 7227	. . . . . . 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 3980	. . 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 28479	. 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 5086  ^◡ccnv 5625   ↾ cres 5628  –1-1-onto→wf1o 6493  ‘cfv 6494  (class class class)co 7362  ωcom 7812   No csur 27621   <s clts 27622   bday cbday 27623   ≤s cles 27726   0_s c0s 27815   +_s cadds 27969   /_su cdivs 28197  ℕ_0scn0s 28322  ℤ_sczs 28388  2_sc2s 28420  ↑_scexps 28422  ℤ_s[1/2]cz12s 28424

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 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-dc 10363  ax-ac2 10380

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-oadd 8404  df-nadd 8597  df-er 8638  df-map 8770  df-en 8889  df-dom 8890  df-fin 8892  df-card 9858  df-acn 9861  df-ac 10033  df-no 27624  df-lts 27625  df-bday 27626  df-les 27727  df-slts 27768  df-cuts 27770  df-0s 27817  df-1s 27818  df-made 27837  df-old 27838  df-left 27840  df-right 27841  df-norec 27948  df-norec2 27959  df-adds 27970  df-negs 28031  df-subs 28032  df-muls 28117  df-divs 28198  df-ons 28262  df-seqs 28294  df-n0s 28324  df-nns 28325  df-zs 28389  df-2s 28421  df-exps 28423  df-z12s 28425

This theorem is referenced by:  bdayfin  28497