Theorem negbdaylem 28475

Description: Lemma for negbday 28476. 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‘( -s‘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 6916	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (bday‘( -s‘𝑥)) = (bday‘( -s‘𝑥𝑂)))
2		fveq2 6911	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (bday‘𝑥) = (bday‘𝑥𝑂))
3	1, 2	sseq12d 3946	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘( -s‘𝑥)) ⊆ (bday‘𝑥) ↔ (bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)))
4		2fveq3 6916	. . 3 ⊢ (𝑥 = 𝐴 → (bday‘( -s‘𝑥)) = (bday‘( -s‘𝐴)))
5		fveq2 6911	. . 3 ⊢ (𝑥 = 𝐴 → (bday‘𝑥) = (bday‘𝐴))
6	4, 5	sseq12d 3946	. 2 ⊢ (𝑥 = 𝐴 → ((bday‘( -s‘𝑥)) ⊆ (bday‘𝑥) ↔ (bday‘( -s‘𝐴)) ⊆ (bday‘𝐴)))
7		negsval 28444	. . . . . 6 ⊢ (𝑥 ∈ No → ( -s‘𝑥) = (( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥))))
8	7	fveq2d 6915	. . . . 5 ⊢ (𝑥 ∈ No → (bday‘( -s‘𝑥)) = (bday‘(( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥)))))
9	8	adantr 486	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → (bday‘( -s‘𝑥)) = (bday‘(( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥)))))
10		negcut2 28459	. . . . 5 ⊢ (𝑥 ∈ No → ( -s “ (R‘𝑥)) <<s ( -s “ (L‘𝑥)))
11		lrold 28316	. . . . . . . . . 10 ⊢ ((L‘𝑥) ∪ (R‘𝑥)) = (O‘(bday‘𝑥))
12		uncom 4160	. . . . . . . . . 10 ⊢ ((L‘𝑥) ∪ (R‘𝑥)) = ((R‘𝑥) ∪ (L‘𝑥))
13	11, 12	eqtr3i 2788	. . . . . . . . 9 ⊢ (O‘(bday‘𝑥)) = ((R‘𝑥) ∪ (L‘𝑥))
14	13	imaeq2i 6063	. . . . . . . 8 ⊢ ( -s “ (O‘(bday‘𝑥))) = ( -s “ ((R‘𝑥) ∪ (L‘𝑥)))
15		imaundi 6154	. . . . . . . 8 ⊢ ( -s “ ((R‘𝑥) ∪ (L‘𝑥))) = (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))
16	14, 15	eqtri 2786	. . . . . . 7 ⊢ ( -s “ (O‘(bday‘𝑥))) = (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))
17	16	imaeq2i 6063	. . . . . 6 ⊢ (bday “ ( -s “ (O‘(bday‘𝑥)))) = (bday “ (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥))))
18	11	raleqi 3321	. . . . . . 7 ⊢ (∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) ↔ ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂))
19		oldbdayim 28308	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 ∈ (O‘(bday‘𝑥)) → (bday‘𝑥𝑂) ∈ (bday‘𝑥))
20	19	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ (O‘(bday‘𝑥))) → (bday‘𝑥𝑂) ∈ (bday‘𝑥))
21		bdayon 28171	. . . . . . . . . . . . 13 ⊢ (bday‘( -s‘𝑥𝑂)) ∈ On
22		bdayon 28171	. . . . . . . . . . . . 13 ⊢ (bday‘𝑥) ∈ On
23		ontr2 6432	. . . . . . . . . . . . 13 ⊢ (((bday‘( -s‘𝑥𝑂)) ∈ On ∧ (bday‘𝑥) ∈ On) → (((bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) ∧ (bday‘𝑥𝑂) ∈ (bday‘𝑥)) → (bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥)))
24	21, 22, 23	mp2an 705	. . . . . . . . . . . 12 ⊢ (((bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) ∧ (bday‘𝑥𝑂) ∈ (bday‘𝑥)) → (bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥))
25	24	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ (O‘(bday‘𝑥))) → (((bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) ∧ (bday‘𝑥𝑂) ∈ (bday‘𝑥)) → (bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥)))
26	20, 25	mpan2d 707	. . . . . . . . . 10 ⊢ ((𝑥 ∈ No ∧ 𝑥𝑂 ∈ (O‘(bday‘𝑥))) → ((bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) → (bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥)))
27	26	ralimdva 3178	. . . . . . . . 9 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) → ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥)))
28	27	imp 412	. . . . . . . 8 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥))
29		bdayfun 28166	. . . . . . . . . 10 ⊢ Fun bday
30		imassrn 6211	. . . . . . . . . . 11 ⊢ ( -s “ (O‘(bday‘𝑥))) ⊆ ran -s
31		bdaydm 28168	. . . . . . . . . . . 12 ⊢ dom bday = No
32		negsfo 28472	. . . . . . . . . . . . 13 ⊢ -s:No⟶ontoNo
33		forn 6824	. . . . . . . . . . . . 13 ⊢ ( -s:No⟶ontoNo → ran -s = No)
34	32, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran -s = No
35	31, 34	eqtr4i 2789	. . . . . . . . . . 11 ⊢ dom bday = ran -s
36	30, 35	sseqtrri 3962	. . . . . . . . . 10 ⊢ ( -s “ (O‘(bday‘𝑥))) ⊆ dom bday
37		funimass4 6975	. . . . . . . . . 10 ⊢ ((Fun bday ∧ ( -s “ (O‘(bday‘𝑥))) ⊆ dom bday) → ((bday “ ( -s “ (O‘(bday‘𝑥)))) ⊆ (bday‘𝑥) ↔ ∀𝑦 ∈ ( -s “ (O‘(bday‘𝑥)))(bday‘𝑦) ∈ (bday‘𝑥)))
38	29, 36, 37	mp2an 705	. . . . . . . . 9 ⊢ ((bday “ ( -s “ (O‘(bday‘𝑥)))) ⊆ (bday‘𝑥) ↔ ∀𝑦 ∈ ( -s “ (O‘(bday‘𝑥)))(bday‘𝑦) ∈ (bday‘𝑥))
39		negsfn 28442	. . . . . . . . . 10 ⊢ -s Fn No
40		oldssno 28260	. . . . . . . . . 10 ⊢ (O‘(bday‘𝑥)) ⊆ No
41		fveq2 6911	. . . . . . . . . . . 12 ⊢ (𝑦 = ( -s‘𝑥𝑂) → (bday‘𝑦) = (bday‘( -s‘𝑥𝑂)))
42	41	eleq1d 2849	. . . . . . . . . . 11 ⊢ (𝑦 = ( -s‘𝑥𝑂) → ((bday‘𝑦) ∈ (bday‘𝑥) ↔ (bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥)))
43	42	ralima 7269	. . . . . . . . . 10 ⊢ (( -s Fn No ∧ (O‘(bday‘𝑥)) ⊆ No) → (∀𝑦 ∈ ( -s “ (O‘(bday‘𝑥)))(bday‘𝑦) ∈ (bday‘𝑥) ↔ ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥)))
44	39, 40, 43	mp2an 705	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ( -s “ (O‘(bday‘𝑥)))(bday‘𝑦) ∈ (bday‘𝑥) ↔ ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥))
45	38, 44	bitri 278	. . . . . . . 8 ⊢ ((bday “ ( -s “ (O‘(bday‘𝑥)))) ⊆ (bday‘𝑥) ↔ ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥))
46	28, 45	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → (bday “ ( -s “ (O‘(bday‘𝑥)))) ⊆ (bday‘𝑥))
47	18, 46	sylan2b 606	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → (bday “ ( -s “ (O‘(bday‘𝑥)))) ⊆ (bday‘𝑥))
48	17, 47	eqsstrrid 3952	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → (bday “ (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))) ⊆ (bday‘𝑥))
49		cutbdaybnd 28214	. . . . . 6 ⊢ ((( -s “ (R‘𝑥)) <<s ( -s “ (L‘𝑥)) ∧ (bday‘𝑥) ∈ On ∧ (bday “ (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))) ⊆ (bday‘𝑥)) → (bday‘(( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥)))) ⊆ (bday‘𝑥))
50	22, 49	mp3an2 1479	. . . . 5 ⊢ ((( -s “ (R‘𝑥)) <<s ( -s “ (L‘𝑥)) ∧ (bday “ (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))) ⊆ (bday‘𝑥)) → (bday‘(( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥)))) ⊆ (bday‘𝑥))
51	10, 48, 50	syl2an2r 698	. . . 4 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → (bday‘(( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥)))) ⊆ (bday‘𝑥))
52	9, 51	eqsstrd 3947	. . 3 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → (bday‘( -s‘𝑥)) ⊆ (bday‘𝑥))
53	52	ex 418	. 2 ⊢ (𝑥 ∈ No → (∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) → (bday‘( -s‘𝑥)) ⊆ (bday‘𝑥)))
54	3, 6, 53	noinds 28364	1 ⊢ (𝐴 ∈ No → (bday‘( -s‘𝐴)) ⊆ (bday‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∪ cun 4152   class class class wbr 5106  dom cdm 5656  ran crn 5657   “ cima 5659  Oncon0 6383  Fun wfun 6555   Fn wfn 6556  ⟶ontowfo 6559  ‘cfv 6561  (class class class)co 7446  Nocsur 28030  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178  Ocold 28242  Lcleft 28244  Rcright 28245   -_scnegs 28438

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440

This theorem is used by:  negbday  28476