Theorem addbdaylem 28087

Description: Lemma for addbday 28088. (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 7400	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → (𝐴 +s 𝑦𝑂) = (𝐴 +s 𝑦𝐿))
2	1	fveq2d 6867	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘(𝐴 +s 𝑦𝑂)) = ( bday ‘(𝐴 +s 𝑦𝐿)))
3		fveq2 6863	. . . . . . . . . 10 ⊢ (𝑦𝑂 = 𝑦𝐿 → ( bday ‘𝑦𝑂) = ( bday ‘𝑦𝐿))
4	3	oveq2d 7408	. . . . . . . . 9 ⊢ (𝑦𝑂 = 𝑦𝐿 → (( bday ‘𝐴) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
5	2, 4	sseq12d 3969	. . . . . . . 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 484	. . . . . . . 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 3932	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
10	9	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ (( L ‘𝐵) ∪ ( R ‘𝐵)))
11	5, 7, 10	rspcdva 3582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)))
12		lrold 27967	. . . . . . . . . . . 12 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) = ( O ‘( bday ‘𝐵))
13	8, 12	sseqtri 3984	. . . . . . . . . . 11 ⊢ 𝑆 ⊆ ( O ‘( bday ‘𝐵))
14	13	sseli 3932	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ ( O ‘( bday ‘𝐵)))
15		oldbdayim 27959	. . . . . . . . . 10 ⊢ (𝑦𝐿 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
17	16	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵))
18		bdayon 27822	. . . . . . . . 9 ⊢ ( bday ‘𝑦𝐿) ∈ On
19		bdayon 27822	. . . . . . . . 9 ⊢ ( bday ‘𝐵) ∈ On
20		bdayon 27822	. . . . . . . . 9 ⊢ ( bday ‘𝐴) ∈ On
21		naddel2 8654	. . . . . . . . 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 1481	. . . . . . . 8 ⊢ (( bday ‘𝑦𝐿) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
23	17, 22	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (( bday ‘𝐴) +no ( bday ‘𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
24		bdayon 27822	. . . . . . . 8 ⊢ ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ On
25		naddcl 8642	. . . . . . . . 9 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On)
26	20, 19, 25	mp2an 702	. . . . . . . 8 ⊢ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On
27		ontr2 6390	. . . . . . . 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 702	. . . . . . 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 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → ( bday ‘(𝐴 +s 𝑦𝐿)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30		fveq2 6863	. . . . . . 7 ⊢ (𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) = ( bday ‘(𝐴 +s 𝑦𝐿)))
31	30	eleq1d 2846	. . . . . 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 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
33	32	rexlimdva 3162	. . . 4 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
34	33	alrimiv 1946	. . 3 ⊢ (𝜑 → ∀𝑤(∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿) → ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
35		eqeq1 2765	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝐴 +s 𝑦𝐿) ↔ 𝑤 = (𝐴 +s 𝑦𝐿)))
36	35	rexbidv 3185	. . . 4 ⊢ (𝑧 = 𝑤 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) ↔ ∃𝑦𝐿 ∈ 𝑆 𝑤 = (𝐴 +s 𝑦𝐿)))
37	36	ralab 3655	. . 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 236	. 2 ⊢ (𝜑 → ∀𝑤 ∈ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ( bday ‘𝑤) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
39		bdayfun 27817	. . 3 ⊢ Fun bday
40		addbdaylem.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
41	40	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝐴 ∈ No )
42		leftssno 27943	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) ⊆ No
43		rightssno 27944	. . . . . . . . . . . 12 ⊢ ( R ‘𝐵) ⊆ No
44	42, 43	unssi 4143	. . . . . . . . . . 11 ⊢ (( L ‘𝐵) ∪ ( R ‘𝐵)) ⊆ No
45	8, 44	sstri 3945	. . . . . . . . . 10 ⊢ 𝑆 ⊆ No
46	45	sseli 3932	. . . . . . . . 9 ⊢ (𝑦𝐿 ∈ 𝑆 → 𝑦𝐿 ∈ No )
47	46	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → 𝑦𝐿 ∈ No )
48	41, 47	addscld 28050	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝐴 +s 𝑦𝐿) ∈ No )
49		eleq1 2849	. . . . . . 7 ⊢ (𝑧 = (𝐴 +s 𝑦𝐿) → (𝑧 ∈ No ↔ (𝐴 +s 𝑦𝐿) ∈ No ))
50	48, 49	syl5ibrcom 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦𝐿 ∈ 𝑆) → (𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No ))
51	50	rexlimdva 3162	. . . . 5 ⊢ (𝜑 → (∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿) → 𝑧 ∈ No ))
52	51	abssdv 4020	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ No )
53		bdaydm 27819	. . . 4 ⊢ dom bday = No
54	52, 53	sseqtrrdi 3977	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)} ⊆ dom bday )
55		funimass4 6927	. . 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 596	. 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 259	1 ⊢ (𝜑 → ( bday “ {𝑧 ∣ ∃𝑦𝐿 ∈ 𝑆 𝑧 = (𝐴 +s 𝑦𝐿)}) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1557   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085   ∪ cun 3902   ⊆ wss 3904  dom cdm 5645   “ cima 5648  Oncon0 6342  Fun wfun 6511  ‘cfv 6517  (class class class)co 7392   +no cnadd 8630   No csur 27681   bday cbday 27683   O cold 27893   L cleft 27895   R cright 27896   +_s cadds 28029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec2 28019  df-adds 28030

This theorem is referenced by:  addbday  28088