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Theorem negsbdaylem 27770
Description: Lemma for negsbday 27771. 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 6896 . . 3 (𝑥 = 𝑥𝑂 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝑥𝑂)))
2 fveq2 6891 . . 3 (𝑥 = 𝑥𝑂 → ( bday 𝑥) = ( bday 𝑥𝑂))
31, 2sseq12d 4015 . 2 (𝑥 = 𝑥𝑂 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)))
4 2fveq3 6896 . . 3 (𝑥 = 𝐴 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝐴)))
5 fveq2 6891 . . 3 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
64, 5sseq12d 4015 . 2 (𝑥 = 𝐴 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴)))
7 negsval 27740 . . . . . 6 (𝑥 No → ( -us𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
87fveq2d 6895 . . . . 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 27754 . . . . 5 (𝑥 No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11 lrold 27629 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday 𝑥))
12 uncom 4153 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1311, 12eqtr3i 2761 . . . . . . . . 9 ( O ‘( bday 𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1413imaeq2i 6057 . . . . . . . 8 ( -us “ ( O ‘( bday 𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15 imaundi 6149 . . . . . . . 8 ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1614, 15eqtri 2759 . . . . . . 7 ( -us “ ( O ‘( bday 𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1716imaeq2i 6057 . . . . . 6 ( bday “ ( -us “ ( O ‘( bday 𝑥)))) = ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥))))
1811raleqi 3322 . . . . . . 7 (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂))
19 oldbdayim 27621 . . . . . . . . . . . 12 (𝑥𝑂 ∈ ( O ‘( bday 𝑥)) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
2019adantl 481 . . . . . . . . . . 11 ((𝑥 No 𝑥𝑂 ∈ ( O ‘( bday 𝑥))) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
21 bdayelon 27515 . . . . . . . . . . . . 13 ( bday ‘( -us𝑥𝑂)) ∈ On
22 bdayelon 27515 . . . . . . . . . . . . 13 ( bday 𝑥) ∈ On
23 ontr2 6411 . . . . . . . . . . . . 13 ((( bday ‘( -us𝑥𝑂)) ∈ On ∧ ( bday 𝑥) ∈ On) → ((( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ∧ ( bday 𝑥𝑂) ∈ ( bday 𝑥)) → ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2421, 22, 23mp2an 689 . . . . . . . . . . . 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 691 . . . . . . . . . 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 27511 . . . . . . . . . 10 Fun bday
30 imassrn 6070 . . . . . . . . . . 11 ( -us “ ( O ‘( bday 𝑥))) ⊆ ran -us
31 bdaydm 27513 . . . . . . . . . . . 12 dom bday = No
32 negsfo 27767 . . . . . . . . . . . . 13 -us : No onto No
33 forn 6808 . . . . . . . . . . . . 13 ( -us : No onto No → ran -us = No )
3432, 33ax-mp 5 . . . . . . . . . . . 12 ran -us = No
3531, 34eqtr4i 2762 . . . . . . . . . . 11 dom bday = ran -us
3630, 35sseqtrri 4019 . . . . . . . . . 10 ( -us “ ( O ‘( bday 𝑥))) ⊆ dom bday
37 funimass4 6956 . . . . . . . . . 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 689 . . . . . . . . 9 (( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday 𝑥)))( bday 𝑦) ∈ ( bday 𝑥))
39 negsfn 27738 . . . . . . . . . 10 -us Fn No
40 oldssno 27594 . . . . . . . . . 10 ( O ‘( bday 𝑥)) ⊆ No
41 fveq2 6891 . . . . . . . . . . . 12 (𝑦 = ( -us𝑥𝑂) → ( bday 𝑦) = ( bday ‘( -us𝑥𝑂)))
4241eleq1d 2817 . . . . . . . . . . 11 (𝑦 = ( -us𝑥𝑂) → (( bday 𝑦) ∈ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
4342imaeqsalv 7364 . . . . . . . . . 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 689 . . . . . . . . 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 233 . . . . . . 7 ((𝑥 No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥))
4718, 46sylan2b 593 . . . . . 6 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥))
4817, 47eqsstrrid 4031 . . . . 5 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday 𝑥))
49 scutbdaybnd 27554 . . . . . 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 1448 . . . . 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 682 . . . 4 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday 𝑥))
529, 51eqsstrd 4020 . . 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 27668 1 (𝐴 No → ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  cun 3946  wss 3948   class class class wbr 5148  dom cdm 5676  ran crn 5677  cima 5679  Oncon0 6364  Fun wfun 6537   Fn wfn 6538  ontowfo 6541  cfv 6543  (class class class)co 7412   No csur 27380   bday cbday 27382   <<s csslt 27519   |s cscut 27521   O cold 27576   L cleft 27578   R cright 27579   -us cnegs 27734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-ot 4637  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-1o 8470  df-2o 8471  df-nadd 8669  df-no 27383  df-slt 27384  df-bday 27385  df-sle 27485  df-sslt 27520  df-scut 27522  df-0s 27563  df-made 27580  df-old 27581  df-left 27583  df-right 27584  df-norec 27661  df-norec2 27672  df-adds 27683  df-negs 27736
This theorem is referenced by:  negsbday  27771
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