Theorem addsbdaylem 28058

Description: Lemma for addsbday 28059. (Contributed by Scott Fenton, 13-Aug-2025.)

Hypotheses

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

Assertion

Ref	Expression
addsbdaylem	⊢ (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 addsbdaylem

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7453	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝐴 +s 𝑦𝑂) = (𝐴 +s 𝑦𝐿))
2	1	fveq2d 6923	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘(𝐴 +s 𝑦𝑂)) = ( bday ‘(𝐴 +s 𝑦𝐿)))
3		fveq2 6919	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘𝑦𝑂) = ( bday ‘𝑦𝐿))
4	3	oveq2d 7461	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
5	2, 4	sseq12d 4036	. . . . . . . 8 ⊢ (𝑦𝑂 = 𝑦𝐿 → (( bday ‘(𝐴 +s 𝑦𝑂)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)) ↔ ( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿))))
6		addsbdaylem.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		addsbdaylem.3	. . . . . . . . . 10 ⊢ 𝑆 ⊆ (( L ‘𝐵) ∪ ( R ‘𝐵))
9	8	sseli 3998	. . . . . . . . 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 3632	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
12		lrold 27944	. . . . . . . . . . . 12 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) = ( O ‘( bday ‘𝐵))
13	8, 12	sseqtri 4039	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ( O ‘( bday ‘𝐵))
14	13	sseli 3998	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ ( O ‘( bday ‘𝐵)))
15		oldbdayim 27936	. . . . . . . . . 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		bdayelon 27830	. . . . . . . . 9 ⊢ ( bday ‘𝑦𝐿) ∈ On
19		bdayelon 27830	. . . . . . . . 9 ⊢ ( bday ‘𝐵) ∈ On
20		bdayelon 27830	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
21		naddel2 8740	. . . . . . . . 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 1461	. . . . . . . 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		bdayelon 27830	. . . . . . . 8 ⊢ ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ On
25		naddcl 8729	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On)
26	20, 19, 25	mp2an 691	. . . . . . . 8 ⊢ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On
27		ontr2 6441	. . . . . . . 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 691	. . . . . . 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 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30		fveq2 6919	. . . . . . 7 ⊢ (𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) = ( bday ‘(𝐴 +s 𝑦𝐿)))
31	30	eleq1d 2823	. . . . . 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 3157	. . . 4 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
34	33	alrimiv 1926	. . 3 ⊢ (𝜑 → ∀𝑤(∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
35		eqeq1 2738	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝐴 +s 𝑦𝐿) ↔ 𝑤 = (𝐴 +s 𝑦𝐿)))
36	35	rexbidv 3181	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) ↔ ∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿)))
37	36	ralab 3707	. . 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 27826	. . 3 ⊢ Fun bday
40		addsbdaylem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
41	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝐴 ∈ No )
42		leftssno 27928	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) ⊆ No
43		rightssno 27929	. . . . . . . . . . . 12 ⊢ ( R ‘𝐵) ⊆ No
44	42, 43	unssi 4208	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
45	8, 44	sstri 4012	. . . . . . . . . 10 ⊢ 𝑆 ⊆ No
46	45	sseli 3998	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ No )
47	46	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ No )
48	41, 47	addscld 28022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝐴 +s 𝑦𝐿) ∈ No )
49		eleq1 2826	. . . . . . 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 3157	. . . . 5 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No ))
52	51	abssdv 4085	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ No )
53		bdaydm 27828	. . . 4 ⊢ dom bday = No
54	52, 53	sseqtrrdi 4054	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ dom bday )
55		funimass4 6985	. . 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 586	. 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 1535   = wceq 1537   ∈ wcel 2103  {cab 2711  ∀wral 3063  ∃wrex 3072   ∪ cun 3968   ⊆ wss 3970  dom cdm 5699   “ cima 5702  Oncon0 6394  Fun wfun 6566  ‘cfv 6572  (class class class)co 7445   +no cnadd 8717   No csur 27693   bday cbday 27695   O cold 27891   L cleft 27893   R cright 27894   +_s cadds 28001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-1o 8518  df-2o 8519  df-nadd 8718  df-no 27696  df-slt 27697  df-bday 27698  df-sslt 27835  df-scut 27837  df-0s 27878  df-made 27895  df-old 27896  df-left 27898  df-right 27899  df-norec2 27991  df-adds 28002

This theorem is referenced by:  addsbday  28059