Theorem addbdaylem 28034

Description: Lemma for addbday 28035. (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 7371	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝐴 +s 𝑦𝑂) = (𝐴 +s 𝑦𝐿))
2	1	fveq2d 6838	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘(𝐴 +s 𝑦𝑂)) = ( bday ‘(𝐴 +s 𝑦𝐿)))
3		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘𝑦𝑂) = ( bday ‘𝑦𝐿))
4	3	oveq2d 7379	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
5	2, 4	sseq12d 3955	. . . . . . . 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 481	. . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
11	5, 7, 10	rspcdva 3568	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
12		lrold 27914	. . . . . . . . . . . 12 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) = ( O ‘( bday ‘𝐵))
13	8, 12	sseqtri 3970	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ( O ‘( bday ‘𝐵))
14	13	sseli 3918	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ ( O ‘( bday ‘𝐵)))
15		oldbdayim 27906	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
18		bdayon 27769	. . . . . . . . 9 ⊢ ( bday ‘𝑦𝐿) ∈ On
19		bdayon 27769	. . . . . . . . 9 ⊢ ( bday ‘𝐵) ∈ On
20		bdayon 27769	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
21		naddel2 8621	. . . . . . . . 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 1469	. . . . . . . 8 ⊢ (( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
23	17, 22	sylib 219	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
24		bdayon 27769	. . . . . . . 8 ⊢ ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ On
25		naddcl 8610	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On)
26	20, 19, 25	mp2an 698	. . . . . . . 8 ⊢ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On
27		ontr2 6365	. . . . . . . 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 698	. . . . . . 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 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30		fveq2 6834	. . . . . . 7 ⊢ (𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) = ( bday ‘(𝐴 +s 𝑦𝐿)))
31	30	eleq1d 2825	. . . . . 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 248	. . . . 5 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
33	32	rexlimdva 3141	. . . 4 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
34	33	alrimiv 1934	. . 3 ⊢ (𝜑 → ∀𝑤(∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
35		eqeq1 2744	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝐴 +s 𝑦𝐿) ↔ 𝑤 = (𝐴 +s 𝑦𝐿)))
36	35	rexbidv 3164	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) ↔ ∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿)))
37	36	ralab 3641	. . 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 235	. 2 ⊢ (𝜑 → ∀𝑤 ∈ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
39		bdayfun 27765	. . 3 ⊢ Fun bday
40		addbdaylem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
41	40	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝐴 ∈ No )
42		leftssno 27890	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) ⊆ No
43		rightssno 27891	. . . . . . . . . . . 12 ⊢ ( R ‘𝐵) ⊆ No
44	42, 43	unssi 4127	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
45	8, 44	sstri 3931	. . . . . . . . . 10 ⊢ 𝑆 ⊆ No
46	45	sseli 3918	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ No )
47	46	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ No )
48	41, 47	addscld 27997	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝐴 +s 𝑦𝐿) ∈ No )
49		eleq1 2828	. . . . . . 7 ⊢ (𝑧 = (𝐴 +s 𝑦𝐿) → (𝑧 ∈ No ↔ (𝐴 +s 𝑦𝐿) ∈ No ))
50	48, 49	syl5ibrcom 248	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No ))
51	50	rexlimdva 3141	. . . . 5 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No ))
52	51	abssdv 4005	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ No )
53		bdaydm 27767	. . . 4 ⊢ dom bday = No
54	52, 53	sseqtrrdi 3963	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ dom bday )
55		funimass4 6898	. . 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 593	. 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 258	1 ⊢ (𝜑 → ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054  ∃wrex 3064   ∪ cun 3888   ⊆ wss 3890  dom cdm 5625   “ cima 5628  Oncon0 6317  Fun wfun 6486  ‘cfv 6492  (class class class)co 7363   +no cnadd 8598   No csur 27628   bday cbday 27630   O cold 27840   L cleft 27842   R cright 27843   +_s cadds 27976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-slts 27775  df-cuts 27777  df-0s 27824  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec2 27966  df-adds 27977

This theorem is referenced by:  addbday  28035