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Theorem negsproplem6 28080
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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
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 → ( -us𝐵) <s ( -us𝐴))
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 27752 . . 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 839 . 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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
8 uncom 4168 . . . . . . . . . 10 (( bday 𝐴) ∪ ( bday 𝐵)) = (( bday 𝐵) ∪ ( bday 𝐴))
98eleq2i 2831 . . . . . . . . 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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))) ↔ ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐵) ∪ ( bday 𝐴)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
11102ralbii 3126 . . . . . . 7 (∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))) ↔ ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐵) ∪ ( bday 𝐴)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
127, 11sylib 218 . . . . . 6 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐵) ∪ ( bday 𝐴)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
1312, 2negsproplem3 28077 . . . . 5 (𝜑 → (( -us𝐵) ∈ No ∧ ( -us “ ( R ‘𝐵)) <<s {( -us𝐵)} ∧ {( -us𝐵)} <<s ( -us “ ( L ‘𝐵))))
1413simp1d 1141 . . . 4 (𝜑 → ( -us𝐵) ∈ No )
1514adantr 480 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝐵) ∈ No )
167adantr 480 . . . . 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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
17 simprl 771 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 No )
18 0sno 27886 . . . . . 6 0s No
1918a1i 11 . . . . 5 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 0s No )
20 bday0s 27888 . . . . . . . 8 ( bday ‘ 0s ) = ∅
2120uneq2i 4175 . . . . . . 7 (( bday 𝑑) ∪ ( bday ‘ 0s )) = (( bday 𝑑) ∪ ∅)
22 un0 4400 . . . . . . 7 (( bday 𝑑) ∪ ∅) = ( bday 𝑑)
2321, 22eqtri 2763 . . . . . 6 (( bday 𝑑) ∪ ( bday ‘ 0s )) = ( bday 𝑑)
24 simprr1 1220 . . . . . . 7 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( bday 𝑑) ∈ ( bday 𝐴))
25 elun1 4192 . . . . . . 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 2843 . . . . 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 28075 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → (( -us𝑑) ∈ No ∧ (𝑑 <s 0s → ( -us ‘ 0s ) <s ( -us𝑑))))
2928simpld 494 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝑑) ∈ No )
307, 1negsproplem3 28077 . . . . 5 (𝜑 → (( -us𝐴) ∈ No ∧ ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)} ∧ {( -us𝐴)} <<s ( -us “ ( L ‘𝐴))))
3130simp1d 1141 . . . 4 (𝜑 → ( -us𝐴) ∈ No )
3231adantr 480 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝐴) ∈ No )
3313simp3d 1143 . . . . 5 (𝜑 → {( -us𝐵)} <<s ( -us “ ( L ‘𝐵)))
3433adantr 480 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → {( -us𝐵)} <<s ( -us “ ( L ‘𝐵)))
35 fvex 6920 . . . . . 6 ( -us𝐵) ∈ V
3635snid 4667 . . . . 5 ( -us𝐵) ∈ {( -us𝐵)}
3736a1i 11 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝐵) ∈ {( -us𝐵)})
38 negsfn 28070 . . . . 5 -us Fn No
39 leftssno 27934 . . . . 5 ( L ‘𝐵) ⊆ No
40 bdayelon 27836 . . . . . . . . 9 ( bday 𝐴) ∈ On
41 oldbday 27954 . . . . . . . . 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 257 . . . . . . 7 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday 𝐴)))
443fveq2d 6911 . . . . . . . 8 (𝜑 → ( O ‘( bday 𝐴)) = ( O ‘( bday 𝐵)))
4544adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( O ‘( bday 𝐴)) = ( O ‘( bday 𝐵)))
4643, 45eleqtrd 2841 . . . . . 6 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 ∈ ( O ‘( bday 𝐵)))
47 simprr3 1222 . . . . . 6 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝑑 <s 𝐵)
48 leftval 27917 . . . . . . 7 ( L ‘𝐵) = {𝑑 ∈ ( O ‘( bday 𝐵)) ∣ 𝑑 <s 𝐵}
4948reqabi 3457 . . . . . 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 7253 . . . . 5 (( -us Fn No ∧ ( L ‘𝐵) ⊆ No 𝑑 ∈ ( L ‘𝐵)) → ( -us𝑑) ∈ ( -us “ ( L ‘𝐵)))
5238, 39, 50, 51mp3an12i 1464 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝑑) ∈ ( -us “ ( L ‘𝐵)))
5334, 37, 52ssltsepcd 27854 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝐵) <s ( -us𝑑))
5430simp2d 1142 . . . . 5 (𝜑 → ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)})
5554adantr 480 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us “ ( R ‘𝐴)) <<s {( -us𝐴)})
56 rightssno 27935 . . . . 5 ( R ‘𝐴) ⊆ No
57 simprr2 1221 . . . . . 6 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → 𝐴 <s 𝑑)
58 rightval 27918 . . . . . . 7 ( R ‘𝐴) = {𝑑 ∈ ( O ‘( bday 𝐴)) ∣ 𝐴 <s 𝑑}
5958reqabi 3457 . . . . . 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 7253 . . . . 5 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No 𝑑 ∈ ( R ‘𝐴)) → ( -us𝑑) ∈ ( -us “ ( R ‘𝐴)))
6238, 56, 60, 61mp3an12i 1464 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝑑) ∈ ( -us “ ( R ‘𝐴)))
63 fvex 6920 . . . . . 6 ( -us𝐴) ∈ V
6463snid 4667 . . . . 5 ( -us𝐴) ∈ {( -us𝐴)}
6564a1i 11 . . . 4 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝐴) ∈ {( -us𝐴)})
6655, 62, 65ssltsepcd 27854 . . 3 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝑑) <s ( -us𝐴))
6715, 29, 32, 53, 66slttrd 27819 . 2 ((𝜑 ∧ (𝑑 No ∧ (( bday 𝑑) ∈ ( bday 𝐴) ∧ 𝐴 <s 𝑑𝑑 <s 𝐵))) → ( -us𝐵) <s ( -us𝐴))
686, 67rexlimddv 3159 1 (𝜑 → ( -us𝐵) <s ( -us𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cun 3961  wss 3963  c0 4339  {csn 4631   class class class wbr 5148  cima 5692  Oncon0 6386   Fn wfn 6558  cfv 6563   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840   0s c0s 27882   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-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-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  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-negs 28068
This theorem is referenced by:  negsproplem7  28081
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