Theorem addbdaylem 27997

Description: Lemma for addbday 27998. (Contributed by Scott Fenton, 13-Aug-2025.)

Hypotheses

Ref	Expression
addbdaylem.1	⊢ (𝜑 → 𝐴 ∈ No )
addbdaylem.2	⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))( bday ‘(𝐴 +s 𝑦𝑂)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)))
addbdaylem.3	⊢ 𝑆 ⊆ (( L ‘𝐵) ∪ ( R ‘𝐵))

Assertion

Ref	Expression
addbdaylem	⊢ (𝜑 → ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

Distinct variable groups:   𝐴,𝑦_𝑂,𝑦_𝐿,𝑧   𝐵,𝑦_𝑂,𝑦_𝐿,𝑧   𝜑,𝑦_𝐿,𝑧   𝑧,𝑆

Allowed substitution hints:   𝜑(𝑦_𝑂)   𝑆(𝑦_𝑂,𝑦_𝐿)

Proof of Theorem addbdaylem

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝐴 +s 𝑦𝑂) = (𝐴 +s 𝑦𝐿))
2	1	fveq2d 6836	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘(𝐴 +s 𝑦𝑂)) = ( bday ‘(𝐴 +s 𝑦𝐿)))
3		fveq2 6832	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘𝑦𝑂) = ( bday ‘𝑦𝐿))
4	3	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
5	2, 4	sseq12d 3956	. . . . . . . 8 ⊢ (𝑦𝑂 = 𝑦𝐿 → (( bday ‘(𝐴 +s 𝑦𝑂)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)) ↔ ( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿))))
6		addbdaylem.2	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))( bday ‘(𝐴 +s 𝑦𝑂)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)))
7	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ∀𝑦𝑂 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵))( bday ‘(𝐴 +s 𝑦𝑂)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)))
8		addbdaylem.3	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (( L ‘𝐵) ∪ ( R ‘𝐵))
9	8	sseli 3918	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
10	9	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
11	5, 7, 10	rspcdva 3566	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
12		lrold 27877	. . . . . . . . . . . 12 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) = ( O ‘( bday ‘𝐵))
13	8, 12	sseqtri 3971	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ( O ‘( bday ‘𝐵))
14	13	sseli 3918	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ ( O ‘( bday ‘𝐵)))
15		oldbdayim 27869	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
18		bdayon 27732	. . . . . . . . 9 ⊢ ( bday ‘𝑦𝐿) ∈ On
19		bdayon 27732	. . . . . . . . 9 ⊢ ( bday ‘𝐵) ∈ On
20		bdayon 27732	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
21		naddel2 8615	. . . . . . . . 9 ⊢ ((( bday ‘𝑦𝐿) ∈ On ∧ ( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
22	18, 19, 20, 21	mp3an 1464	. . . . . . . 8 ⊢ (( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
23	17, 22	sylib 218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
24		bdayon 27732	. . . . . . . 8 ⊢ ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ On
25		naddcl 8604	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On)
26	20, 19, 25	mp2an 693	. . . . . . . 8 ⊢ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On
27		ontr2 6363	. . . . . . . 8 ⊢ ((( bday ‘(𝐴 +s 𝑦𝐿)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) → ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
28	24, 26, 27	mp2an 693	. . . . . . 7 ⊢ ((( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) → ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
29	11, 23, 28	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30		fveq2 6832	. . . . . . 7 ⊢ (𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) = ( bday ‘(𝐴 +s 𝑦𝐿)))
31	30	eleq1d 2822	. . . . . 6 ⊢ (𝑤 = (𝐴 +s 𝑦𝐿) → (( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
32	29, 31	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
33	32	rexlimdva 3139	. . . 4 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
34	33	alrimiv 1929	. . 3 ⊢ (𝜑 → ∀𝑤(∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
35		eqeq1 2741	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝐴 +s 𝑦𝐿) ↔ 𝑤 = (𝐴 +s 𝑦𝐿)))
36	35	rexbidv 3162	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) ↔ ∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿)))
37	36	ralab 3640	. . 3 ⊢ (∀𝑤 ∈ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ∀𝑤(∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
38	34, 37	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑤 ∈ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
39		bdayfun 27728	. . 3 ⊢ Fun bday
40		addbdaylem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
41	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝐴 ∈ No )
42		leftssno 27853	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) ⊆ No
43		rightssno 27854	. . . . . . . . . . . 12 ⊢ ( R ‘𝐵) ⊆ No
44	42, 43	unssi 4132	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
45	8, 44	sstri 3932	. . . . . . . . . 10 ⊢ 𝑆 ⊆ No
46	45	sseli 3918	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ No )
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ No )
48	41, 47	addscld 27960	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝐴 +s 𝑦𝐿) ∈ No )
49		eleq1 2825	. . . . . . 7 ⊢ (𝑧 = (𝐴 +s 𝑦𝐿) → (𝑧 ∈ No ↔ (𝐴 +s 𝑦𝐿) ∈ No ))
50	48, 49	syl5ibrcom 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No ))
51	50	rexlimdva 3139	. . . . 5 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No ))
52	51	abssdv 4008	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ No )
53		bdaydm 27730	. . . 4 ⊢ dom bday = No
54	52, 53	sseqtrrdi 3964	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ dom bday )
55		funimass4 6896	. . 3 ⊢ ((Fun bday ∧ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ dom bday ) → (( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
56	39, 54, 55	sylancr 588	. 2 ⊢ (𝜑 → (( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ∀𝑤 ∈ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
57	38, 56	mpbird 257	1 ⊢ (𝜑 → ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∪ cun 3888   ⊆ wss 3890  dom cdm 5622   “ cima 5625  Oncon0 6315  Fun wfun 6484  ‘cfv 6490  (class class class)co 7358   +no cnadd 8592   No csur 27591   bday cbday 27593   O cold 27803   L cleft 27805   R cright 27806   +_s cadds 27939

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-1o 8396  df-2o 8397  df-nadd 8593  df-no 27594  df-lts 27595  df-bday 27596  df-slts 27738  df-cuts 27740  df-0s 27787  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec2 27929  df-adds 27940

This theorem is referenced by:  addbday  27998