Theorem negsbdaylem 27967

Description: Lemma for negsbday 27968. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.)

Assertion

Ref	Expression
negsbdaylem	⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))

Proof of Theorem negsbdaylem

Dummy variables 𝑥 𝑥_𝑂 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6827	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝑥𝑂)))
2		fveq2 6822	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘𝑥) = ( bday ‘𝑥𝑂))
3	1, 2	sseq12d 3969	. 2 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)))
4		2fveq3 6827	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝐴)))
5		fveq2 6822	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
6	4, 5	sseq12d 3969	. 2 ⊢ (𝑥 = 𝐴 → (( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴)))
7		negsval 27936	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us ‘𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
8	7	fveq2d 6826	. . . . 5 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
9	8	adantr 480	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday ‘( -us ‘𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
10		negscut2 27951	. . . . 5 ⊢ (𝑥 ∈ No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11		lrold 27811	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday ‘𝑥))
12		uncom 4109	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
13	11, 12	eqtr3i 2754	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
14	13	imaeq2i 6009	. . . . . . . 8 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15		imaundi 6098	. . . . . . . 8 ⊢ ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
16	14, 15	eqtri 2752	. . . . . . 7 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
17	16	imaeq2i 6009	. . . . . 6 ⊢ ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) = ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥))))
18	11	raleqi 3287	. . . . . . 7 ⊢ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂))
19		oldbdayim 27803	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 ∈ ( O ‘( bday ‘𝑥)) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
20	19	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
21		bdayelon 27686	. . . . . . . . . . . . 13 ⊢ ( bday ‘( -us ‘𝑥𝑂)) ∈ On
22		bdayelon 27686	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑥) ∈ On
23		ontr2 6355	. . . . . . . . . . . . 13 ⊢ ((( bday ‘( -us ‘𝑥𝑂)) ∈ On ∧ ( bday ‘𝑥) ∈ On) → ((( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ∧ ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥)) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
24	21, 22, 23	mp2an 692	. . . . . . . . . . . 12 ⊢ ((( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ∧ ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥)) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → ((( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ∧ ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥)) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
26	20, 25	mpan2d 694	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → (( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
27	26	ralimdva 3141	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
28	27	imp 406	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
29		bdayfun 27682	. . . . . . . . . 10 ⊢ Fun bday
30		imassrn 6022	. . . . . . . . . . 11 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ ran -us
31		bdaydm 27684	. . . . . . . . . . . 12 ⊢ dom bday = No
32		negsfo 27964	. . . . . . . . . . . . 13 ⊢ -us : No –onto→ No
33		forn 6739	. . . . . . . . . . . . 13 ⊢ ( -us : No –onto→ No → ran -us = No )
34	32, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran -us = No
35	31, 34	eqtr4i 2755	. . . . . . . . . . 11 ⊢ dom bday = ran -us
36	30, 35	sseqtrri 3985	. . . . . . . . . 10 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ dom bday
37		funimass4 6887	. . . . . . . . . 10 ⊢ ((Fun bday ∧ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ dom bday ) → (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥)))
38	29, 36, 37	mp2an 692	. . . . . . . . 9 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥))
39		negsfn 27934	. . . . . . . . . 10 ⊢ -us Fn No
40		oldssno 27771	. . . . . . . . . 10 ⊢ ( O ‘( bday ‘𝑥)) ⊆ No
41		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → ( bday ‘𝑦) = ( bday ‘( -us ‘𝑥𝑂)))
42	41	eleq1d 2813	. . . . . . . . . . 11 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → (( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
43	42	ralima 7173	. . . . . . . . . 10 ⊢ (( -us Fn No ∧ ( O ‘( bday ‘𝑥)) ⊆ No ) → (∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
44	39, 40, 43	mp2an 692	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
45	38, 44	bitri 275	. . . . . . . 8 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
46	28, 45	sylibr 234	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥))
47	18, 46	sylan2b 594	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥))
48	17, 47	eqsstrrid 3975	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
49		scutbdaybnd 27726	. . . . . 6 ⊢ ((( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)) ∧ ( bday ‘𝑥) ∈ On ∧ ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
50	22, 49	mp3an2 1451	. . . . 5 ⊢ ((( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)) ∧ ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
51	10, 48, 50	syl2an2r 685	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
52	9, 51	eqsstrd 3970	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥))
53	52	ex 412	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥)))
54	3, 6, 53	noinds 27857	1 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∪ cun 3901   ⊆ wss 3903   class class class wbr 5092  dom cdm 5619  ran crn 5620   “ cima 5622  Oncon0 6307  Fun wfun 6476   Fn wfn 6477  –onto→wfo 6480  ‘cfv 6482  (class class class)co 7349   No csur 27549   bday cbday 27551   <<s csslt 27691   |s cscut 27693   O cold 27753   L cleft 27755   R cright 27756   -u_s cnegs 27930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-1o 8388  df-2o 8389  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27738  df-made 27757  df-old 27758  df-left 27760  df-right 27761  df-norec 27850  df-norec2 27861  df-adds 27872  df-negs 27932

This theorem is referenced by:  negsbday  27968