MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  negsbdaylem Structured version   Visualization version   GIF version

Theorem negsbdaylem 28103
Description: Lemma for negsbday 28104. 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 6912 . . 3 (𝑥 = 𝑥𝑂 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝑥𝑂)))
2 fveq2 6907 . . 3 (𝑥 = 𝑥𝑂 → ( bday 𝑥) = ( bday 𝑥𝑂))
31, 2sseq12d 4029 . 2 (𝑥 = 𝑥𝑂 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)))
4 2fveq3 6912 . . 3 (𝑥 = 𝐴 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝐴)))
5 fveq2 6907 . . 3 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
64, 5sseq12d 4029 . 2 (𝑥 = 𝐴 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴)))
7 negsval 28072 . . . . . 6 (𝑥 No → ( -us𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
87fveq2d 6911 . . . . 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 28087 . . . . 5 (𝑥 No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11 lrold 27950 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday 𝑥))
12 uncom 4168 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1311, 12eqtr3i 2765 . . . . . . . . 9 ( O ‘( bday 𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1413imaeq2i 6078 . . . . . . . 8 ( -us “ ( O ‘( bday 𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15 imaundi 6172 . . . . . . . 8 ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1614, 15eqtri 2763 . . . . . . 7 ( -us “ ( O ‘( bday 𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1716imaeq2i 6078 . . . . . 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 27942 . . . . . . . . . . . 12 (𝑥𝑂 ∈ ( O ‘( bday 𝑥)) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
2019adantl 481 . . . . . . . . . . 11 ((𝑥 No 𝑥𝑂 ∈ ( O ‘( bday 𝑥))) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
21 bdayelon 27836 . . . . . . . . . . . . 13 ( bday ‘( -us𝑥𝑂)) ∈ On
22 bdayelon 27836 . . . . . . . . . . . . 13 ( bday 𝑥) ∈ On
23 ontr2 6433 . . . . . . . . . . . . 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 3165 . . . . . . . . 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 27832 . . . . . . . . . 10 Fun bday
30 imassrn 6091 . . . . . . . . . . 11 ( -us “ ( O ‘( bday 𝑥))) ⊆ ran -us
31 bdaydm 27834 . . . . . . . . . . . 12 dom bday = No
32 negsfo 28100 . . . . . . . . . . . . 13 -us : No onto No
33 forn 6824 . . . . . . . . . . . . 13 ( -us : No onto No → ran -us = No )
3432, 33ax-mp 5 . . . . . . . . . . . 12 ran -us = No
3531, 34eqtr4i 2766 . . . . . . . . . . 11 dom bday = ran -us
3630, 35sseqtrri 4033 . . . . . . . . . 10 ( -us “ ( O ‘( bday 𝑥))) ⊆ dom bday
37 funimass4 6973 . . . . . . . . . 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 28070 . . . . . . . . . 10 -us Fn No
40 oldssno 27915 . . . . . . . . . 10 ( O ‘( bday 𝑥)) ⊆ No
41 fveq2 6907 . . . . . . . . . . . 12 (𝑦 = ( -us𝑥𝑂) → ( bday 𝑦) = ( bday ‘( -us𝑥𝑂)))
4241eleq1d 2824 . . . . . . . . . . 11 (𝑦 = ( -us𝑥𝑂) → (( bday 𝑦) ∈ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
4342ralima 7257 . . . . . . . . . 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 4045 . . . . 5 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday 𝑥))
49 scutbdaybnd 27875 . . . . . 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 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 4034 . . 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 27993 1 (𝐴 No → ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cun 3961  wss 3963   class class class wbr 5148  dom cdm 5689  ran crn 5690  cima 5692  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  ontowfo 6561  cfv 6563  (class class class)co 7431   No csur 27699   bday cbday 27701   <<s csslt 27840   |s cscut 27842   O cold 27897   L cleft 27899   R cright 27900   -us cnegs 28066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068
This theorem is referenced by:  negsbday  28104
  Copyright terms: Public domain W3C validator