Theorem negbdaylem 28073

Description: Lemma for negbday 28074. Bound the birthday of the negative of a surreal number above. (Contributed by Scott Fenton, 8-Mar-2025.)

Assertion

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

Proof of Theorem negbdaylem

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

Step	Hyp	Ref	Expression
1		2fveq3 6839	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝑥𝑂)))
2		fveq2 6834	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘𝑥) = ( bday ‘𝑥𝑂))
3	1, 2	sseq12d 3955	. 2 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)))
4		2fveq3 6839	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝐴)))
5		fveq2 6834	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
6	4, 5	sseq12d 3955	. 2 ⊢ (𝑥 = 𝐴 → (( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴)))
7		negsval 28042	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us ‘𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
8	7	fveq2d 6838	. . . . 5 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
9	8	adantr 481	. . . 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		negcut2 28057	. . . . 5 ⊢ (𝑥 ∈ No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11		lrold 27914	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday ‘𝑥))
12		uncom 4095	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
13	11, 12	eqtr3i 2765	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
14	13	imaeq2i 6017	. . . . . . . 8 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15		imaundi 6107	. . . . . . . 8 ⊢ ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
16	14, 15	eqtri 2763	. . . . . . 7 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
17	16	imaeq2i 6017	. . . . . 6 ⊢ ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) = ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥))))
18	11	raleqi 3296	. . . . . . 7 ⊢ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂))
19		oldbdayim 27906	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 ∈ ( O ‘( bday ‘𝑥)) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
20	19	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
21		bdayon 27769	. . . . . . . . . . . . 13 ⊢ ( bday ‘( -us ‘𝑥𝑂)) ∈ On
22		bdayon 27769	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑥) ∈ On
23		ontr2 6365	. . . . . . . . . . . . 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 698	. . . . . . . . . . . 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 700	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → (( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
27	26	ralimdva 3152	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
28	27	imp 407	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
29		bdayfun 27765	. . . . . . . . . 10 ⊢ Fun bday
30		imassrn 6030	. . . . . . . . . . 11 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ ran -us
31		bdaydm 27767	. . . . . . . . . . . 12 ⊢ dom bday = No
32		negsfo 28070	. . . . . . . . . . . . 13 ⊢ -us : No –onto→ No
33		forn 6749	. . . . . . . . . . . . 13 ⊢ ( -us : No –onto→ No → ran -us = No )
34	32, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran -us = No
35	31, 34	eqtr4i 2766	. . . . . . . . . . 11 ⊢ dom bday = ran -us
36	30, 35	sseqtrri 3971	. . . . . . . . . 10 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ dom bday
37		funimass4 6898	. . . . . . . . . 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 698	. . . . . . . . 9 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥))
39		negsfn 28040	. . . . . . . . . 10 ⊢ -us Fn No
40		oldssno 27858	. . . . . . . . . 10 ⊢ ( O ‘( bday ‘𝑥)) ⊆ No
41		fveq2 6834	. . . . . . . . . . . 12 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → ( bday ‘𝑦) = ( bday ‘( -us ‘𝑥𝑂)))
42	41	eleq1d 2825	. . . . . . . . . . 11 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → (( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
43	42	ralima 7188	. . . . . . . . . 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 698	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
45	38, 44	bitri 276	. . . . . . . 8 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
46	28, 45	sylibr 235	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥))
47	18, 46	sylan2b 600	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥))
48	17, 47	eqsstrrid 3961	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday ‘𝑥))
49		cutbdaybnd 27812	. . . . . 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 1457	. . . . 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 691	. . . 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 3956	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥))
53	52	ex 413	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥)))
54	3, 6, 53	noinds 27962	1 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ∪ cun 3888   ⊆ wss 3890   class class class wbr 5079  dom cdm 5625  ran crn 5626   “ cima 5628  Oncon0 6317  Fun wfun 6486   Fn wfn 6487  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7363   No csur 27628   bday cbday 27630   <<s cslts 27774   |s ccuts 27776   O cold 27840   L cleft 27842   R cright 27843   -u_s cnegs 28036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038

This theorem is referenced by:  negbday  28074