Theorem negbdaylem 28126

Description: Lemma for negbday 28127. 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 6868	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝑥𝑂)))
2		fveq2 6863	. . 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 6868	. . 3 ⊢ (𝑥 = 𝐴 → ( bday ‘( -us ‘𝑥)) = ( bday ‘( -us ‘𝐴)))
5		fveq2 6863	. . 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 28095	. . . . . 6 ⊢ (𝑥 ∈ No → ( -us ‘𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
8	7	fveq2d 6867	. . . . 5 ⊢ (𝑥 ∈ No → ( bday ‘( -us ‘𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
9	8	adantr 484	. . . 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 28110	. . . . 5 ⊢ (𝑥 ∈ No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11		lrold 27967	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday ‘𝑥))
12		uncom 4111	. . . . . . . . . 10 ⊢ (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
13	11, 12	eqtr3i 2786	. . . . . . . . 9 ⊢ ( O ‘( bday ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
14	13	imaeq2i 6044	. . . . . . . 8 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15		imaundi 6131	. . . . . . . 8 ⊢ ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
16	14, 15	eqtri 2784	. . . . . . 7 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
17	16	imaeq2i 6044	. . . . . 6 ⊢ ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) = ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥))))
18	11	raleqi 3317	. . . . . . 7 ⊢ (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂))
19		oldbdayim 27959	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 ∈ ( O ‘( bday ‘𝑥)) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
20	19	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → ( bday ‘𝑥𝑂) ∈ ( bday ‘𝑥))
21		bdayon 27822	. . . . . . . . . . . . 13 ⊢ ( bday ‘( -us ‘𝑥𝑂)) ∈ On
22		bdayon 27822	. . . . . . . . . . . . 13 ⊢ ( bday ‘𝑥) ∈ On
23		ontr2 6390	. . . . . . . . . . . . 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 702	. . . . . . . . . . . 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 704	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))) → (( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
27	26	ralimdva 3173	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
28	27	imp 410	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
29		bdayfun 27817	. . . . . . . . . 10 ⊢ Fun bday
30		imassrn 6057	. . . . . . . . . . 11 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ ran -us
31		bdaydm 27819	. . . . . . . . . . . 12 ⊢ dom bday = No
32		negsfo 28123	. . . . . . . . . . . . 13 ⊢ -us : No –onto→ No
33		forn 6777	. . . . . . . . . . . . 13 ⊢ ( -us : No –onto→ No → ran -us = No )
34	32, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran -us = No
35	31, 34	eqtr4i 2787	. . . . . . . . . . 11 ⊢ dom bday = ran -us
36	30, 35	sseqtrri 3985	. . . . . . . . . 10 ⊢ ( -us “ ( O ‘( bday ‘𝑥))) ⊆ dom bday
37		funimass4 6927	. . . . . . . . . 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 702	. . . . . . . . 9 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥))
39		negsfn 28093	. . . . . . . . . 10 ⊢ -us Fn No
40		oldssno 27911	. . . . . . . . . 10 ⊢ ( O ‘( bday ‘𝑥)) ⊆ No
41		fveq2 6863	. . . . . . . . . . . 12 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → ( bday ‘𝑦) = ( bday ‘( -us ‘𝑥𝑂)))
42	41	eleq1d 2846	. . . . . . . . . . 11 ⊢ (𝑦 = ( -us ‘𝑥𝑂) → (( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥)))
43	42	ralima 7217	. . . . . . . . . 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 702	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ( -us “ ( O ‘( bday ‘𝑥)))( bday ‘𝑦) ∈ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
45	38, 44	bitri 277	. . . . . . . 8 ⊢ (( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ∈ ( bday ‘𝑥))
46	28, 45	sylibr 236	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday ‘𝑥)))) ⊆ ( bday ‘𝑥))
47	18, 46	sylan2b 603	. . . . . 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		cutbdaybnd 27865	. . . . . 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 1469	. . . . 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 695	. . . 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 416	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us ‘𝑥𝑂)) ⊆ ( bday ‘𝑥𝑂) → ( bday ‘( -us ‘𝑥)) ⊆ ( bday ‘𝑥)))
54	3, 6, 53	noinds 28015	1 ⊢ (𝐴 ∈ No → ( bday ‘( -us ‘𝐴)) ⊆ ( bday ‘𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∪ cun 3902   ⊆ wss 3904   class class class wbr 5099  dom cdm 5645  ran crn 5646   “ cima 5648  Oncon0 6342  Fun wfun 6511   Fn wfn 6512  –onto→wfo 6515  ‘cfv 6517  (class class class)co 7392   No csur 27681   bday cbday 27683   <<s cslts 27827   |s ccuts 27829   O cold 27893   L cleft 27895   R cright 27896   -u_s cnegs 28089

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-ot 4590  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-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091

This theorem is referenced by:  negbday  28127