Theorem negbdaylem 28448

Description: Lemma for negbday 28449. 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 6891	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (bday‘( -s‘𝑥)) = (bday‘( -s‘𝑥𝑂)))
2		fveq2 6886	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (bday‘𝑥) = (bday‘𝑥𝑂))
3	1, 2	sseq12d 3941	. 2 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘( -s‘𝑥)) ⊆ (bday‘𝑥) ↔ (bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)))
4		2fveq3 6891	. . 3 ⊢ (𝑥 = 𝐴 → (bday‘( -s‘𝑥)) = (bday‘( -s‘𝐴)))
5		fveq2 6886	. . 3 ⊢ (𝑥 = 𝐴 → (bday‘𝑥) = (bday‘𝐴))
6	4, 5	sseq12d 3941	. 2 ⊢ (𝑥 = 𝐴 → ((bday‘( -s‘𝑥)) ⊆ (bday‘𝑥) ↔ (bday‘( -s‘𝐴)) ⊆ (bday‘𝐴)))
7		negsval 28417	. . . . . 6 ⊢ (𝑥 ∈ No → ( -s‘𝑥) = (( -s “ (R‘𝑥)) |s ( -s “ (L‘𝑥))))
8	7	fveq2d 6890	. . . . 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 28432	. . . . 5 ⊢ (𝑥 ∈ No → ( -s “ (R‘𝑥)) <<s ( -s “ (L‘𝑥)))
11		lrold 28289	. . . . . . . . . 10 ⊢ ((L‘𝑥) ∪ (R‘𝑥)) = (O‘(bday‘𝑥))
12		uncom 4157	. . . . . . . . . 10 ⊢ ((L‘𝑥) ∪ (R‘𝑥)) = ((R‘𝑥) ∪ (L‘𝑥))
13	11, 12	eqtr3i 2785	. . . . . . . . 9 ⊢ (O‘(bday‘𝑥)) = ((R‘𝑥) ∪ (L‘𝑥))
14	13	imaeq2i 6049	. . . . . . . 8 ⊢ ( -s “ (O‘(bday‘𝑥))) = ( -s “ ((R‘𝑥) ∪ (L‘𝑥)))
15		imaundi 6140	. . . . . . . 8 ⊢ ( -s “ ((R‘𝑥) ∪ (L‘𝑥))) = (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))
16	14, 15	eqtri 2783	. . . . . . 7 ⊢ ( -s “ (O‘(bday‘𝑥))) = (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))
17	16	imaeq2i 6049	. . . . . 6 ⊢ (bday “ ( -s “ (O‘(bday‘𝑥)))) = (bday “ (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥))))
18	11	raleqi 3317	. . . . . . 7 ⊢ (∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂) ↔ ∀𝑥𝑂 ∈ (O‘(bday‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂))
19		oldbdayim 28281	. . . . . . . . . . . 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 28144	. . . . . . . . . . . . 13 ⊢ (bday‘( -s‘𝑥𝑂)) ∈ On
22		bdayon 28144	. . . . . . . . . . . . 13 ⊢ (bday‘𝑥) ∈ On
23		ontr2 6411	. . . . . . . . . . . . 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 3174	. . . . . . . . 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 28139	. . . . . . . . . 10 ⊢ Fun bday
30		imassrn 6197	. . . . . . . . . . 11 ⊢ ( -s “ (O‘(bday‘𝑥))) ⊆ ran -s
31		bdaydm 28141	. . . . . . . . . . . 12 ⊢ dom bday = No
32		negsfo 28445	. . . . . . . . . . . . 13 ⊢ -s:No–onto→No
33		forn 6800	. . . . . . . . . . . . 13 ⊢ ( -s:No–onto→No → ran -s = No)
34	32, 33	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran -s = No
35	31, 34	eqtr4i 2786	. . . . . . . . . . 11 ⊢ dom bday = ran -s
36	30, 35	sseqtrri 3957	. . . . . . . . . 10 ⊢ ( -s “ (O‘(bday‘𝑥))) ⊆ dom bday
37		funimass4 6950	. . . . . . . . . 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 28415	. . . . . . . . . 10 ⊢ -s Fn No
40		oldssno 28233	. . . . . . . . . 10 ⊢ (O‘(bday‘𝑥)) ⊆ No
41		fveq2 6886	. . . . . . . . . . . 12 ⊢ (𝑦 = ( -s‘𝑥𝑂) → (bday‘𝑦) = (bday‘( -s‘𝑥𝑂)))
42	41	eleq1d 2845	. . . . . . . . . . 11 ⊢ (𝑦 = ( -s‘𝑥𝑂) → ((bday‘𝑦) ∈ (bday‘𝑥) ↔ (bday‘( -s‘𝑥𝑂)) ∈ (bday‘𝑥)))
43	42	ralima 7244	. . . . . . . . . 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 3947	. . . . 5 ⊢ ((𝑥 ∈ No ∧ ∀𝑥𝑂 ∈ ((L‘𝑥) ∪ (R‘𝑥))(bday‘( -s‘𝑥𝑂)) ⊆ (bday‘𝑥𝑂)) → (bday “ (( -s “ (R‘𝑥)) ∪ ( -s “ (L‘𝑥)))) ⊆ (bday‘𝑥))
49		cutbdaybnd 28187	. . . . . 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 1478	. . . . 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 3942	. . 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 28337	1 ⊢ (𝐴 ∈ No → (bday‘( -s‘𝐴)) ⊆ (bday‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∪ cun 4149   class class class wbr 5102  dom cdm 5650  ran crn 5651   “ cima 5653  Oncon0 6362  Fun wfun 6533   Fn wfn 6534  –onto→wfo 6537  ‘cfv 6539  (class class class)co 7421  Nocsur 28003  bdaycbday 28005   <<s cslts 28149   |s ccuts 28151  Ocold 28215  Lcleft 28217  Rcright 28218   -_scnegs 28411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-norec2 28341  df-adds 28352  df-negs 28413

This theorem is used by:  negbday  28449