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Theorem negsbdaylem 28089
Description: Lemma for negsbday 28090. 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 ‘( -us𝐴)) ⊆ ( bday 𝐴))

Proof of Theorem negsbdaylem
Dummy variables 𝑥 𝑥𝑂 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 2fveq3 6910 . . 3 (𝑥 = 𝑥𝑂 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝑥𝑂)))
2 fveq2 6905 . . 3 (𝑥 = 𝑥𝑂 → ( bday 𝑥) = ( bday 𝑥𝑂))
31, 2sseq12d 4016 . 2 (𝑥 = 𝑥𝑂 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)))
4 2fveq3 6910 . . 3 (𝑥 = 𝐴 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝐴)))
5 fveq2 6905 . . 3 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
64, 5sseq12d 4016 . 2 (𝑥 = 𝐴 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴)))
7 negsval 28058 . . . . . 6 (𝑥 No → ( -us𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
87fveq2d 6909 . . . . 5 (𝑥 No → ( bday ‘( -us𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
98adantr 480 . . . 4 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday ‘( -us𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
10 negscut2 28073 . . . . 5 (𝑥 No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11 lrold 27936 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday 𝑥))
12 uncom 4157 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1311, 12eqtr3i 2766 . . . . . . . . 9 ( O ‘( bday 𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1413imaeq2i 6075 . . . . . . . 8 ( -us “ ( O ‘( bday 𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15 imaundi 6168 . . . . . . . 8 ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1614, 15eqtri 2764 . . . . . . 7 ( -us “ ( O ‘( bday 𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1716imaeq2i 6075 . . . . . 6 ( bday “ ( -us “ ( O ‘( bday 𝑥)))) = ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥))))
1811raleqi 3323 . . . . . . 7 (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂))
19 oldbdayim 27928 . . . . . . . . . . . 12 (𝑥𝑂 ∈ ( O ‘( bday 𝑥)) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
2019adantl 481 . . . . . . . . . . 11 ((𝑥 No 𝑥𝑂 ∈ ( O ‘( bday 𝑥))) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
21 bdayelon 27822 . . . . . . . . . . . . 13 ( bday ‘( -us𝑥𝑂)) ∈ On
22 bdayelon 27822 . . . . . . . . . . . . 13 ( bday 𝑥) ∈ On
23 ontr2 6430 . . . . . . . . . . . . 13 ((( bday ‘( -us𝑥𝑂)) ∈ On ∧ ( bday 𝑥) ∈ On) → ((( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ∧ ( bday 𝑥𝑂) ∈ ( bday 𝑥)) → ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2421, 22, 23mp2an 692 . . . . . . . . . . . 12 ((( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ∧ ( bday 𝑥𝑂) ∈ ( bday 𝑥)) → ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥))
2524a1i 11 . . . . . . . . . . 11 ((𝑥 No 𝑥𝑂 ∈ ( O ‘( bday 𝑥))) → ((( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ∧ ( bday 𝑥𝑂) ∈ ( bday 𝑥)) → ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2620, 25mpan2d 694 . . . . . . . . . 10 ((𝑥 No 𝑥𝑂 ∈ ( O ‘( bday 𝑥))) → (( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) → ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2726ralimdva 3166 . . . . . . . . 9 (𝑥 No → (∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) → ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2827imp 406 . . . . . . . 8 ((𝑥 No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥))
29 bdayfun 27818 . . . . . . . . . 10 Fun bday
30 imassrn 6088 . . . . . . . . . . 11 ( -us “ ( O ‘( bday 𝑥))) ⊆ ran -us
31 bdaydm 27820 . . . . . . . . . . . 12 dom bday = No
32 negsfo 28086 . . . . . . . . . . . . 13 -us : No onto No
33 forn 6822 . . . . . . . . . . . . 13 ( -us : No onto No → ran -us = No )
3432, 33ax-mp 5 . . . . . . . . . . . 12 ran -us = No
3531, 34eqtr4i 2767 . . . . . . . . . . 11 dom bday = ran -us
3630, 35sseqtrri 4032 . . . . . . . . . 10 ( -us “ ( O ‘( bday 𝑥))) ⊆ dom bday
37 funimass4 6972 . . . . . . . . . 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)))
3829, 36, 37mp2an 692 . . . . . . . . 9 (( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday 𝑥)))( bday 𝑦) ∈ ( bday 𝑥))
39 negsfn 28056 . . . . . . . . . 10 -us Fn No
40 oldssno 27901 . . . . . . . . . 10 ( O ‘( bday 𝑥)) ⊆ No
41 fveq2 6905 . . . . . . . . . . . 12 (𝑦 = ( -us𝑥𝑂) → ( bday 𝑦) = ( bday ‘( -us𝑥𝑂)))
4241eleq1d 2825 . . . . . . . . . . 11 (𝑦 = ( -us𝑥𝑂) → (( bday 𝑦) ∈ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
4342ralima 7258 . . . . . . . . . 10 (( -us Fn No ∧ ( O ‘( bday 𝑥)) ⊆ No ) → (∀𝑦 ∈ ( -us “ ( O ‘( bday 𝑥)))( bday 𝑦) ∈ ( bday 𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
4439, 40, 43mp2an 692 . . . . . . . . 9 (∀𝑦 ∈ ( -us “ ( O ‘( bday 𝑥)))( bday 𝑦) ∈ ( bday 𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥))
4538, 44bitri 275 . . . . . . . 8 (( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥))
4628, 45sylibr 234 . . . . . . 7 ((𝑥 No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥))
4718, 46sylan2b 594 . . . . . 6 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥))
4817, 47eqsstrrid 4022 . . . . 5 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday 𝑥))
49 scutbdaybnd 27861 . . . . . 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))
5022, 49mp3an2 1450 . . . . 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))
5110, 48, 50syl2an2r 685 . . . 4 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday 𝑥))
529, 51eqsstrd 4017 . . 3 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday ‘( -us𝑥)) ⊆ ( bday 𝑥))
5352ex 412 . 2 (𝑥 No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) → ( bday ‘( -us𝑥)) ⊆ ( bday 𝑥)))
543, 6, 53noinds 27979 1 (𝐴 No → ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  cun 3948  wss 3950   class class class wbr 5142  dom cdm 5684  ran crn 5685  cima 5687  Oncon0 6383  Fun wfun 6554   Fn wfn 6555  ontowfo 6558  cfv 6560  (class class class)co 7432   No csur 27685   bday cbday 27687   <<s csslt 27826   |s cscut 27828   O cold 27883   L cleft 27885   R cright 27886   -us cnegs 28052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054
This theorem is referenced by:  negsbday  28090
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