Theorem addbdaylem 28433

Description: Lemma for addbday 28434. (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 7451	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝐴 +s 𝑦𝑂) = (𝐴 +s 𝑦𝐿))
2	1	fveq2d 6912	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → (bday‘(𝐴 +s 𝑦𝑂)) = (bday‘(𝐴 +s 𝑦𝐿)))
3		fveq2 6908	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (bday‘𝑦𝑂) = (bday‘𝑦𝐿))
4	3	oveq2d 7459	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → ((bday‘𝐴) +no (bday‘𝑦𝑂)) = ((bday‘𝐴) +no (bday‘𝑦𝐿)))
5	2, 4	sseq12d 3946	. . . . . . . 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 486	. . . . . . . 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 3909	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
10	9	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ ((L‘𝐵) ∪ (R‘𝐵)))
11	5, 7, 10	rspcdva 3581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (bday‘(𝐴 +s 𝑦𝐿)) ⊆ ((bday‘𝐴) +no (bday‘𝑦𝐿)))
12		lrold 28313	. . . . . . . . . . . 12 ⊢ ((L‘𝐵) ∪ (R‘𝐵)) = (O‘(bday‘𝐵))
13	8, 12	sseqtri 3961	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ (O‘(bday‘𝐵))
14	13	sseli 3909	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ (O‘(bday‘𝐵)))
15		oldbdayim 28305	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ (O‘(bday‘𝐵)) → (bday‘𝑦𝐿) ∈ (bday‘𝐵))
16	14, 15	syl 18	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → (bday‘𝑦𝐿) ∈ (bday‘𝐵))
17	16	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (bday‘𝑦𝐿) ∈ (bday‘𝐵))
18		bdayon 28168	. . . . . . . . 9 ⊢ (bday‘𝑦𝐿) ∈ On
19		bdayon 28168	. . . . . . . . 9 ⊢ (bday‘𝐵) ∈ On
20		bdayon 28168	. . . . . . . . 9 ⊢ (bday‘𝐴) ∈ On
21		naddel2 8726	. . . . . . . . 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 1491	. . . . . . . 8 ⊢ ((bday‘𝑦𝐿) ∈ (bday‘𝐵) ↔ ((bday‘𝐴) +no (bday‘𝑦𝐿)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
23	17, 22	sylib 221	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ((bday‘𝐴) +no (bday‘𝑦𝐿)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
24		bdayon 28168	. . . . . . . 8 ⊢ (bday‘(𝐴 +s 𝑦𝐿)) ∈ On
25		naddcl 8714	. . . . . . . . 9 ⊢ (((bday‘𝐴) ∈ On ∧ (bday‘𝐵) ∈ On) → ((bday‘𝐴) +no (bday‘𝐵)) ∈ On)
26	20, 19, 25	mp2an 705	. . . . . . . 8 ⊢ ((bday‘𝐴) +no (bday‘𝐵)) ∈ On
27		ontr2 6429	. . . . . . . 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 705	. . . . . . 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 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (bday‘(𝐴 +s 𝑦𝐿)) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
30		fveq2 6908	. . . . . . 7 ⊢ (𝑤 = (𝐴 +s 𝑦𝐿) → (bday‘𝑤) = (bday‘(𝐴 +s 𝑦𝐿)))
31	30	eleq1d 2849	. . . . . 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 250	. . . . 5 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑤 = (𝐴 +s 𝑦𝐿) → (bday‘𝑤) ∈ ((bday‘𝐴) +no (bday‘𝐵))))
33	32	rexlimdva 3167	. . . 4 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → (bday‘𝑤) ∈ ((bday‘𝐴) +no (bday‘𝐵))))
34	33	alrimiv 1962	. . 3 ⊢ (𝜑 → ∀𝑤(∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → (bday‘𝑤) ∈ ((bday‘𝐴) +no (bday‘𝐵))))
35		eqeq1 2767	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝐴 +s 𝑦𝐿) ↔ 𝑤 = (𝐴 +s 𝑦𝐿)))
36	35	rexbidv 3190	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) ↔ ∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿)))
37	36	ralab 3654	. . 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 237	. 2 ⊢ (𝜑 → ∀𝑤 ∈ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} (bday‘𝑤) ∈ ((bday‘𝐴) +no (bday‘𝐵)))
39		bdayfun 28163	. . 3 ⊢ Fun bday
40		addbdaylem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No)
41	40	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝐴 ∈ No)
42		leftssno 28289	. . . . . . . . . . . 12 ⊢ (L‘𝐵) ⊆ No
43		rightssno 28290	. . . . . . . . . . . 12 ⊢ (R‘𝐵) ⊆ No
44	42, 43	unssi 4192	. . . . . . . . . . 11 ⊢ ((L‘𝐵) ∪ (R‘𝐵)) ⊆ No
45	8, 44	sstri 3922	. . . . . . . . . 10 ⊢ 𝑆 ⊆ No
46	45	sseli 3909	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ No)
47	46	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ No)
48	41, 47	addscld 28396	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝐴 +s 𝑦𝐿) ∈ No)
49		eleq1 2852	. . . . . . 7 ⊢ (𝑧 = (𝐴 +s 𝑦𝐿) → (𝑧 ∈ No ↔ (𝐴 +s 𝑦𝐿) ∈ No))
50	48, 49	syl5ibrcom 250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No))
51	50	rexlimdva 3167	. . . . 5 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No))
52	51	abssdv 3995	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ No)
53		bdaydm 28165	. . . 4 ⊢ dom bday = No
54	52, 53	sseqtrrdi 3954	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ dom bday)
55		funimass4 6972	. . 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 599	. 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 260	1 ⊢ (𝜑 → (bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ ((bday‘𝐴) +no (bday‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∪ cun 4152  dom cdm 5655   “ cima 5658  Oncon0 6380  Fun wfun 6552  ‘cfv 6558  (class class class)co 7443   +no cnadd 8702  Nocsur 28027  bdaycbday 28029  Ocold 28239  Lcleft 28241  Rcright 28242   +_s cadds 28375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176  df-0s 28223  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec2 28365  df-adds 28376

This theorem is used by:  addbday  28434