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Theorem negsproplem6 34319
Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is the same age as 𝐵. (Contributed by Scott Fenton, 3-Feb-2025.)
Hypotheses
Ref Expression
negsproplem.1 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
negsproplem4.1 (𝜑𝐴 No )
negsproplem4.2 (𝜑𝐵 No )
negsproplem4.3 (𝜑𝐴 <s 𝐵)
negsproplem6.4 (𝜑 → ( bday 𝐴) = ( bday 𝐵))
Assertion
Ref Expression
negsproplem6 (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem6
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 negsproplem4.1 . . 3 (𝜑𝐴 No )
2 negsproplem4.2 . . 3 (𝜑𝐵 No )
3 negsproplem6.4 . . 3 (𝜑 → ( bday 𝐴) = ( bday 𝐵))
4 negsproplem4.3 . . 3 (𝜑𝐴 <s 𝐵)
5 nodense 26991 . . 3 (((𝐴 No 𝐵 No ) ∧ (( bday 𝐴) = ( bday 𝐵) ∧ 𝐴 <s 𝐵)) → ∃𝑑 No (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))
61, 2, 3, 4, 5syl22anc 837 . 2 (𝜑 → ∃𝑑 No (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))
7 negsproplem.1 . . . . . . 7 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
8 uncom 4111 . . . . . . . . . 10 (( bday 𝐴) ∪ ( bday 𝐵)) = (( bday 𝐵) ∪ ( bday 𝐴))
98eleq2i 2829 . . . . . . . . 9 ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) ↔ (( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐵) ∪ ( bday 𝐴)))
109imbi1i 349 . . . . . . . 8 (((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐵) ∪ ( bday 𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
11102ralbii 3125 . . . . . . 7 (∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))) ↔ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐵) ∪ ( bday 𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
127, 11sylib 217 . . . . . 6 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐵) ∪ ( bday 𝐴)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
1312, 2negsproplem3 34316 . . . . 5 (𝜑 → (( -us ‘𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us ‘𝐵)} ∧ {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵))))
1413simp1d 1142 . . . 4 (𝜑 → ( -us ‘𝐵) ∈ No )
1514adantr 481 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝐵) ∈ No )
167adantr 481 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
17 simprl 769 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 No )
18 0sno 27116 . . . . . 6 0s ∈ No
1918a1i 11 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 0s ∈ No )
20 bday0s 27118 . . . . . . . 8 ( bday ‘ 0s ) = ∅
2120uneq2i 4118 . . . . . . 7 (( bday 𝑑) ∪ ( bday ‘ 0s )) = (( bday 𝑑) ∪ ∅)
22 un0 4348 . . . . . . 7 (( bday 𝑑) ∪ ∅) = ( bday 𝑑)
2321, 22eqtri 2765 . . . . . 6 (( bday 𝑑) ∪ ( bday ‘ 0s )) = ( bday 𝑑)
24 simprr1 1221 . . . . . . 7 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( bday 𝑑) ∈ ( bday 𝐴))
25 elun1 4134 . . . . . . 7 (( bday 𝑑) ∈ ( bday 𝐴) → ( bday 𝑑) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2624, 25syl 17 . . . . . 6 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( bday 𝑑) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2723, 26eqeltrid 2842 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → (( bday 𝑑) ∪ ( bday ‘ 0s )) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2816, 17, 19, 27negsproplem1 34314 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → (( -us ‘𝑑) ∈ No ∧ (𝑑 <s 0s → ( -us ‘ 0s ) <s ( -us ‘𝑑))))
2928simpld 495 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ No )
307, 1negsproplem3 34316 . . . . 5 (𝜑 → (( -us ‘𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)} ∧ {( -us ‘𝐴)} <<s ( -us “ ( L ‘𝐴))))
3130simp1d 1142 . . . 4 (𝜑 → ( -us ‘𝐴) ∈ No )
3231adantr 481 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝐴) ∈ No )
3313simp3d 1144 . . . . 5 (𝜑 → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
3433adantr 481 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → {( -us ‘𝐵)} <<s ( -us “ ( L ‘𝐵)))
35 fvex 6852 . . . . . 6 ( -us ‘𝐵) ∈ V
3635snid 4620 . . . . 5 ( -us ‘𝐵) ∈ {( -us ‘𝐵)}
3736a1i 11 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝐵) ∈ {( -us ‘𝐵)})
38 negsfn 34310 . . . . 5 -us Fn No
39 leftssno 27160 . . . . 5 ( L ‘𝐵) ⊆ No
40 bdayelon 27067 . . . . . . . . 9 ( bday 𝐴) ∈ On
41 oldbday 27178 . . . . . . . . 9 ((( bday 𝐴) ∈ On ∧ 𝑑 No ) → (𝑑 ∈ ( O ‘( bday 𝐴)) ↔ ( bday 𝑑) ∈ ( bday 𝐴)))
4240, 17, 41sylancr 587 . . . . . . . 8 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → (𝑑 ∈ ( O ‘( bday 𝐴)) ↔ ( bday 𝑑) ∈ ( bday 𝐴)))
4324, 42mpbird 256 . . . . . . 7 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday 𝐴)))
443fveq2d 6843 . . . . . . . 8 (𝜑 → ( O ‘( bday 𝐴)) = ( O ‘( bday 𝐵)))
4544adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( O ‘( bday 𝐴)) = ( O ‘( bday 𝐵)))
4643, 45eleqtrd 2840 . . . . . 6 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday 𝐵)))
47 simprr3 1223 . . . . . 6 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 <s 𝐵)
48 leftval 27144 . . . . . . 7 ( L ‘𝐵) = {𝑑 ∈ ( O ‘( bday 𝐵)) ∣ 𝑑 <s 𝐵}
4948rabeq2i 3427 . . . . . 6 (𝑑 ∈ ( L ‘𝐵) ↔ (𝑑 ∈ ( O ‘( bday 𝐵)) ∧ 𝑑 <s 𝐵))
5046, 47, 49sylanbrc 583 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 ∈ ( L ‘𝐵))
51 fnfvima 7179 . . . . 5 (( -us Fn No ∧ ( L ‘𝐵) ⊆ No 𝑑 ∈ ( L ‘𝐵)) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
5238, 39, 50, 51mp3an12i 1465 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( L ‘𝐵)))
5334, 37, 52ssltsepcd 27084 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝑑))
5430simp2d 1143 . . . . 5 (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
5554adantr 481 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us “ ( R ‘𝐴)) <<s {( -us ‘𝐴)})
56 rightssno 27161 . . . . 5 ( R ‘𝐴) ⊆ No
57 simprr2 1222 . . . . . 6 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝐴 <s 𝑑)
58 rightval 27145 . . . . . . 7 ( R ‘𝐴) = {𝑑 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑑}
5958rabeq2i 3427 . . . . . 6 (𝑑 ∈ ( R ‘𝐴) ↔ (𝑑 ∈ ( O ‘( bday 𝐴)) ∧ 𝐴 <s 𝑑))
6043, 57, 59sylanbrc 583 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 ∈ ( R ‘𝐴))
61 fnfvima 7179 . . . . 5 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No 𝑑 ∈ ( R ‘𝐴)) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
6238, 56, 60, 61mp3an12i 1465 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝑑) ∈ ( -us “ ( R ‘𝐴)))
63 fvex 6852 . . . . . 6 ( -us ‘𝐴) ∈ V
6463snid 4620 . . . . 5 ( -us ‘𝐴) ∈ {( -us ‘𝐴)}
6564a1i 11 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝐴) ∈ {( -us ‘𝐴)})
6655, 62, 65ssltsepcd 27084 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝑑) <s ( -us ‘𝐴))
6715, 29, 32, 53, 66slttrd 27058 . 2 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us ‘𝐵) <s ( -us ‘𝐴))
686, 67rexlimddv 3156 1 (𝜑 → ( -us ‘𝐵) <s ( -us ‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wrex 3071  cun 3906  wss 3908  c0 4280  {csn 4584   class class class wbr 5103  cima 5634  Oncon0 6315   Fn wfn 6488  cfv 6493   No csur 26939   <s cslt 26940   bday cbday 26941   <<s csslt 27071   0s c0s 27112   O cold 27124   L cleft 27126   R cright 27127   -us cnegs 34306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-int 4906  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-se 5587  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-1o 8404  df-2o 8405  df-no 26942  df-slt 26943  df-bday 26944  df-sslt 27072  df-scut 27074  df-0s 27114  df-made 27128  df-old 27129  df-left 27131  df-right 27132  df-norec 34246  df-negs 34308
This theorem is referenced by:  negsproplem7  34320
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