Theorem addbdaylem 28408

Description: Lemma for addbday 28409. (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 7428	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝐴 +s 𝑦𝑂) = (𝐴 +s 𝑦𝐿))
2	1	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → (bday‘(𝐴 +s 𝑦𝑂)) = (bday‘(𝐴 +s 𝑦𝐿)))
3		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (bday‘𝑦𝑂) = (bday‘𝑦𝐿))
4	3	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → ((bday‘𝐴) +no (bday‘𝑦𝑂)) = ((bday‘𝐴) +no (bday‘𝑦𝐿)))
5	2, 4	sseq12d 3941	. . . . . . . 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 3904	. . . . . . . . 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 3577	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (bday‘(𝐴 +s 𝑦𝐿)) ⊆ ((bday‘𝐴) +no (bday‘𝑦𝐿)))
12		lrold 28288	. . . . . . . . . . . 12 ⊢ ((L‘𝐵) ∪ (R‘𝐵)) = (O‘(bday‘𝐵))
13	8, 12	sseqtri 3956	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ (O‘(bday‘𝐵))
14	13	sseli 3904	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ (O‘(bday‘𝐵)))
15		oldbdayim 28280	. . . . . . . . . 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 28143	. . . . . . . . 9 ⊢ (bday‘𝑦𝐿) ∈ On
19		bdayon 28143	. . . . . . . . 9 ⊢ (bday‘𝐵) ∈ On
20		bdayon 28143	. . . . . . . . 9 ⊢ (bday‘𝐴) ∈ On
21		naddel2 8701	. . . . . . . . 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 1490	. . . . . . . 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 28143	. . . . . . . 8 ⊢ (bday‘(𝐴 +s 𝑦𝐿)) ∈ On
25		naddcl 8689	. . . . . . . . 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 6411	. . . . . . . 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 6885	. . . . . . 7 ⊢ (𝑤 = (𝐴 +s 𝑦𝐿) → (bday‘𝑤) = (bday‘(𝐴 +s 𝑦𝐿)))
31	30	eleq1d 2845	. . . . . 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 3163	. . . 4 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → (bday‘𝑤) ∈ ((bday‘𝐴) +no (bday‘𝐵))))
34	33	alrimiv 1959	. . 3 ⊢ (𝜑 → ∀𝑤(∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → (bday‘𝑤) ∈ ((bday‘𝐴) +no (bday‘𝐵))))
35		eqeq1 2764	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝐴 +s 𝑦𝐿) ↔ 𝑤 = (𝐴 +s 𝑦𝐿)))
36	35	rexbidv 3186	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) ↔ ∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿)))
37	36	ralab 3650	. . 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 28138	. . 3 ⊢ Fun bday
40		addbdaylem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No)
41	40	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝐴 ∈ No)
42		leftssno 28264	. . . . . . . . . . . 12 ⊢ (L‘𝐵) ⊆ No
43		rightssno 28265	. . . . . . . . . . . 12 ⊢ (R‘𝐵) ⊆ No
44	42, 43	unssi 4189	. . . . . . . . . . 11 ⊢ ((L‘𝐵) ∪ (R‘𝐵)) ⊆ No
45	8, 44	sstri 3917	. . . . . . . . . 10 ⊢ 𝑆 ⊆ No
46	45	sseli 3904	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ No)
47	46	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ No)
48	41, 47	addscld 28371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝐴 +s 𝑦𝐿) ∈ No)
49		eleq1 2848	. . . . . . 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 3163	. . . . 5 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No))
52	51	abssdv 3992	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ No)
53		bdaydm 28140	. . . 4 ⊢ dom bday = No
54	52, 53	sseqtrrdi 3949	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ dom bday)
55		funimass4 6949	. . 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 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∪ cun 4149  dom cdm 5650   “ cima 5653  Oncon0 6362  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420   +no cnadd 8677  Nocsur 28002  bdaycbday 28004  Ocold 28214  Lcleft 28216  Rcright 28217   +_s cadds 28350

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec2 28340  df-adds 28351

This theorem is used by:  addbday  28409