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Theorem negsbdaylem 27510
Description: Lemma for negsbday 27511. 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 6893 . . 3 (𝑥 = 𝑥𝑂 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝑥𝑂)))
2 fveq2 6888 . . 3 (𝑥 = 𝑥𝑂 → ( bday 𝑥) = ( bday 𝑥𝑂))
31, 2sseq12d 4014 . 2 (𝑥 = 𝑥𝑂 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)))
4 2fveq3 6893 . . 3 (𝑥 = 𝐴 → ( bday ‘( -us𝑥)) = ( bday ‘( -us𝐴)))
5 fveq2 6888 . . 3 (𝑥 = 𝐴 → ( bday 𝑥) = ( bday 𝐴))
64, 5sseq12d 4014 . 2 (𝑥 = 𝐴 → (( bday ‘( -us𝑥)) ⊆ ( bday 𝑥) ↔ ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴)))
7 negsval 27480 . . . . . 6 (𝑥 No → ( -us𝑥) = (( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥))))
87fveq2d 6892 . . . . 5 (𝑥 No → ( bday ‘( -us𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
98adantr 482 . . . 4 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday ‘( -us𝑥)) = ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))))
10 negscut2 27494 . . . . 5 (𝑥 No → ( -us “ ( R ‘𝑥)) <<s ( -us “ ( L ‘𝑥)))
11 lrold 27371 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = ( O ‘( bday 𝑥))
12 uncom 4152 . . . . . . . . . 10 (( L ‘𝑥) ∪ ( R ‘𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1311, 12eqtr3i 2763 . . . . . . . . 9 ( O ‘( bday 𝑥)) = (( R ‘𝑥) ∪ ( L ‘𝑥))
1413imaeq2i 6055 . . . . . . . 8 ( -us “ ( O ‘( bday 𝑥))) = ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥)))
15 imaundi 6146 . . . . . . . 8 ( -us “ (( R ‘𝑥) ∪ ( L ‘𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1614, 15eqtri 2761 . . . . . . 7 ( -us “ ( O ‘( bday 𝑥))) = (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))
1716imaeq2i 6055 . . . . . 6 ( bday “ ( -us “ ( O ‘( bday 𝑥)))) = ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥))))
1811raleqi 3324 . . . . . . 7 (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ↔ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂))
19 oldbdayim 27363 . . . . . . . . . . . 12 (𝑥𝑂 ∈ ( O ‘( bday 𝑥)) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
2019adantl 483 . . . . . . . . . . 11 ((𝑥 No 𝑥𝑂 ∈ ( O ‘( bday 𝑥))) → ( bday 𝑥𝑂) ∈ ( bday 𝑥))
21 bdayelon 27258 . . . . . . . . . . . . 13 ( bday ‘( -us𝑥𝑂)) ∈ On
22 bdayelon 27258 . . . . . . . . . . . . 13 ( bday 𝑥) ∈ On
23 ontr2 6408 . . . . . . . . . . . . 13 ((( bday ‘( -us𝑥𝑂)) ∈ On ∧ ( bday 𝑥) ∈ On) → ((( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) ∧ ( bday 𝑥𝑂) ∈ ( bday 𝑥)) → ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2421, 22, 23mp2an 691 . . . . . . . . . . . 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 693 . . . . . . . . . 10 ((𝑥 No 𝑥𝑂 ∈ ( O ‘( bday 𝑥))) → (( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) → ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2726ralimdva 3168 . . . . . . . . 9 (𝑥 No → (∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) → ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
2827imp 408 . . . . . . . 8 ((𝑥 No ∧ ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ∀𝑥𝑂 ∈ ( O ‘( bday 𝑥))( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥))
29 bdayfun 27254 . . . . . . . . . 10 Fun bday
30 imassrn 6068 . . . . . . . . . . 11 ( -us “ ( O ‘( bday 𝑥))) ⊆ ran -us
31 bdaydm 27256 . . . . . . . . . . . 12 dom bday = No
32 negsfo 27507 . . . . . . . . . . . . 13 -us : No onto No
33 forn 6805 . . . . . . . . . . . . 13 ( -us : No onto No → ran -us = No )
3432, 33ax-mp 5 . . . . . . . . . . . 12 ran -us = No
3531, 34eqtr4i 2764 . . . . . . . . . . 11 dom bday = ran -us
3630, 35sseqtrri 4018 . . . . . . . . . 10 ( -us “ ( O ‘( bday 𝑥))) ⊆ dom bday
37 funimass4 6953 . . . . . . . . . 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 691 . . . . . . . . 9 (( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥) ↔ ∀𝑦 ∈ ( -us “ ( O ‘( bday 𝑥)))( bday 𝑦) ∈ ( bday 𝑥))
39 negsfn 27478 . . . . . . . . . 10 -us Fn No
40 oldssno 27336 . . . . . . . . . 10 ( O ‘( bday 𝑥)) ⊆ No
41 fveq2 6888 . . . . . . . . . . . 12 (𝑦 = ( -us𝑥𝑂) → ( bday 𝑦) = ( bday ‘( -us𝑥𝑂)))
4241eleq1d 2819 . . . . . . . . . . 11 (𝑦 = ( -us𝑥𝑂) → (( bday 𝑦) ∈ ( bday 𝑥) ↔ ( bday ‘( -us𝑥𝑂)) ∈ ( bday 𝑥)))
4342imaeqsalv 7356 . . . . . . . . . 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 691 . . . . . . . . 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 595 . . . . . 6 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ ( -us “ ( O ‘( bday 𝑥)))) ⊆ ( bday 𝑥))
4817, 47eqsstrrid 4030 . . . . 5 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday “ (( -us “ ( R ‘𝑥)) ∪ ( -us “ ( L ‘𝑥)))) ⊆ ( bday 𝑥))
49 scutbdaybnd 27296 . . . . . 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 684 . . . 4 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday ‘(( -us “ ( R ‘𝑥)) |s ( -us “ ( L ‘𝑥)))) ⊆ ( bday 𝑥))
529, 51eqsstrd 4019 . . 3 ((𝑥 No ∧ ∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂)) → ( bday ‘( -us𝑥)) ⊆ ( bday 𝑥))
5352ex 414 . 2 (𝑥 No → (∀𝑥𝑂 ∈ (( L ‘𝑥) ∪ ( R ‘𝑥))( bday ‘( -us𝑥𝑂)) ⊆ ( bday 𝑥𝑂) → ( bday ‘( -us𝑥)) ⊆ ( bday 𝑥)))
543, 6, 53noinds 27409 1 (𝐴 No → ( bday ‘( -us𝐴)) ⊆ ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  cun 3945  wss 3947   class class class wbr 5147  dom cdm 5675  ran crn 5676  cima 5678  Oncon0 6361  Fun wfun 6534   Fn wfn 6535  ontowfo 6538  cfv 6540  (class class class)co 7404   No csur 27123   bday cbday 27125   <<s csslt 27262   |s cscut 27264   O cold 27318   L cleft 27320   R cright 27321   -us cnegs 27474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-ot 4636  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-1o 8461  df-2o 8462  df-nadd 8661  df-no 27126  df-slt 27127  df-bday 27128  df-sle 27228  df-sslt 27263  df-scut 27265  df-0s 27305  df-made 27322  df-old 27323  df-left 27325  df-right 27326  df-norec 27402  df-norec2 27413  df-adds 27424  df-negs 27476
This theorem is referenced by:  negsbday  27511
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