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Theorem negsproplem2 34315
Description: Lemma for surreal negation. Show that the cut that defines negation is legitimate. (Contributed by Scott Fenton, 2-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)))))
negsproplem2.1 (𝜑𝐴 No )
Assertion
Ref Expression
negsproplem2 (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem negsproplem2
Dummy variables 𝑎 𝑏 𝑥L 𝑥R are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negsfn 34310 . . . 4 -us Fn No
2 fnfun 6599 . . . 4 ( -us Fn No → Fun -us )
31, 2ax-mp 5 . . 3 Fun -us
4 fvex 6852 . . . 4 ( R ‘𝐴) ∈ V
54funimaex 6586 . . 3 (Fun -us → ( -us “ ( R ‘𝐴)) ∈ V)
63, 5mp1i 13 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) ∈ V)
7 fvex 6852 . . . 4 ( L ‘𝐴) ∈ V
87funimaex 6586 . . 3 (Fun -us → ( -us “ ( L ‘𝐴)) ∈ V)
93, 8mp1i 13 . 2 (𝜑 → ( -us “ ( L ‘𝐴)) ∈ V)
10 rightssold 27159 . . . 4 ( R ‘𝐴) ⊆ ( O ‘( bday 𝐴))
11 imass2 6052 . . . 4 (( R ‘𝐴) ⊆ ( O ‘( bday 𝐴)) → ( -us “ ( R ‘𝐴)) ⊆ ( -us “ ( O ‘( bday 𝐴))))
1210, 11ax-mp 5 . . 3 ( -us “ ( R ‘𝐴)) ⊆ ( -us “ ( O ‘( bday 𝐴)))
13 negsproplem.1 . . . . . . . 8 (𝜑 → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
1413adantr 481 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
15 oldssno 27142 . . . . . . . . 9 ( O ‘( bday 𝐴)) ⊆ No
1615sseli 3938 . . . . . . . 8 (𝑎 ∈ ( O ‘( bday 𝐴)) → 𝑎 No )
1716adantl 482 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → 𝑎 No )
18 0sno 27116 . . . . . . . 8 0s ∈ No
1918a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → 0s ∈ No )
20 bday0s 27118 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
2120uneq2i 4118 . . . . . . . . 9 (( bday 𝑎) ∪ ( bday ‘ 0s )) = (( bday 𝑎) ∪ ∅)
22 un0 4348 . . . . . . . . 9 (( bday 𝑎) ∪ ∅) = ( bday 𝑎)
2321, 22eqtri 2765 . . . . . . . 8 (( bday 𝑎) ∪ ( bday ‘ 0s )) = ( bday 𝑎)
24 oldbdayim 27168 . . . . . . . . . 10 (𝑎 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑎) ∈ ( bday 𝐴))
2524adantl 482 . . . . . . . . 9 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ( bday 𝑎) ∈ ( bday 𝐴))
26 elun1 4134 . . . . . . . . 9 (( bday 𝑎) ∈ ( bday 𝐴) → ( bday 𝑎) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2725, 26syl 17 . . . . . . . 8 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ( bday 𝑎) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2823, 27eqeltrid 2842 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → (( bday 𝑎) ∪ ( bday ‘ 0s )) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2914, 17, 19, 28negsproplem1 34314 . . . . . 6 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → (( -us ‘𝑎) ∈ No ∧ (𝑎 <s 0s → ( -us ‘ 0s ) <s ( -us ‘𝑎))))
3029simpld 495 . . . . 5 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ( -us ‘𝑎) ∈ No )
3130ralrimiva 3141 . . . 4 (𝜑 → ∀𝑎 ∈ ( O ‘( bday 𝐴))( -us ‘𝑎) ∈ No )
321fndmi 6603 . . . . . 6 dom -us = No
3315, 32sseqtrri 3979 . . . . 5 ( O ‘( bday 𝐴)) ⊆ dom -us
34 funimass4 6904 . . . . 5 ((Fun -us ∧ ( O ‘( bday 𝐴)) ⊆ dom -us ) → (( -us “ ( O ‘( bday 𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday 𝐴))( -us ‘𝑎) ∈ No ))
353, 33, 34mp2an 690 . . . 4 (( -us “ ( O ‘( bday 𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday 𝐴))( -us ‘𝑎) ∈ No )
3631, 35sylibr 233 . . 3 (𝜑 → ( -us “ ( O ‘( bday 𝐴))) ⊆ No )
3712, 36sstrid 3953 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) ⊆ No )
38 leftssold 27158 . . . 4 ( L ‘𝐴) ⊆ ( O ‘( bday 𝐴))
39 imass2 6052 . . . 4 (( L ‘𝐴) ⊆ ( O ‘( bday 𝐴)) → ( -us “ ( L ‘𝐴)) ⊆ ( -us “ ( O ‘( bday 𝐴))))
4038, 39ax-mp 5 . . 3 ( -us “ ( L ‘𝐴)) ⊆ ( -us “ ( O ‘( bday 𝐴)))
4140, 36sstrid 3953 . 2 (𝜑 → ( -us “ ( L ‘𝐴)) ⊆ No )
42 rightssno 27161 . . . . . . 7 ( R ‘𝐴) ⊆ No
43 fvelimab 6911 . . . . . . 7 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥R ∈ ( R ‘𝐴)( -us ‘𝑥R) = 𝑎))
441, 42, 43mp2an 690 . . . . . 6 (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥R ∈ ( R ‘𝐴)( -us ‘𝑥R) = 𝑎)
45 leftssno 27160 . . . . . . 7 ( L ‘𝐴) ⊆ No
46 fvelimab 6911 . . . . . . 7 (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥L ∈ ( L ‘𝐴)( -us ‘𝑥L) = 𝑏))
471, 45, 46mp2an 690 . . . . . 6 (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥L ∈ ( L ‘𝐴)( -us ‘𝑥L) = 𝑏)
4844, 47anbi12i 627 . . . . 5 ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ (∃𝑥R ∈ ( R ‘𝐴)( -us ‘𝑥R) = 𝑎 ∧ ∃𝑥L ∈ ( L ‘𝐴)( -us ‘𝑥L) = 𝑏))
49 reeanv 3215 . . . . 5 (∃𝑥R ∈ ( R ‘𝐴)∃𝑥L ∈ ( L ‘𝐴)(( -us ‘𝑥R) = 𝑎 ∧ ( -us ‘𝑥L) = 𝑏) ↔ (∃𝑥R ∈ ( R ‘𝐴)( -us ‘𝑥R) = 𝑎 ∧ ∃𝑥L ∈ ( L ‘𝐴)( -us ‘𝑥L) = 𝑏))
5048, 49bitr4i 277 . . . 4 ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ ∃𝑥R ∈ ( R ‘𝐴)∃𝑥L ∈ ( L ‘𝐴)(( -us ‘𝑥R) = 𝑎 ∧ ( -us ‘𝑥L) = 𝑏))
51 negsproplem2.1 . . . . . . . . . 10 (𝜑𝐴 No )
52 lltropt 27153 . . . . . . . . . 10 (𝐴 No → ( L ‘𝐴) <<s ( R ‘𝐴))
5351, 52syl 17 . . . . . . . . 9 (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
5453adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → ( L ‘𝐴) <<s ( R ‘𝐴))
55 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → 𝑥L ∈ ( L ‘𝐴))
56 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → 𝑥R ∈ ( R ‘𝐴))
5754, 55, 56ssltsepcd 27084 . . . . . . 7 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → 𝑥L <s 𝑥R)
5813adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us ‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us ‘𝑦) <s ( -us ‘𝑥)))))
5945sseli 3938 . . . . . . . . . 10 (𝑥L ∈ ( L ‘𝐴) → 𝑥L No )
6059ad2antll 727 . . . . . . . . 9 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → 𝑥L No )
6142sseli 3938 . . . . . . . . . . 11 (𝑥R ∈ ( R ‘𝐴) → 𝑥R No )
6261adantr 481 . . . . . . . . . 10 ((𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴)) → 𝑥R No )
6362adantl 482 . . . . . . . . 9 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → 𝑥R No )
6438sseli 3938 . . . . . . . . . . . . 13 (𝑥L ∈ ( L ‘𝐴) → 𝑥L ∈ ( O ‘( bday 𝐴)))
6564ad2antll 727 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → 𝑥L ∈ ( O ‘( bday 𝐴)))
66 oldbdayim 27168 . . . . . . . . . . . 12 (𝑥L ∈ ( O ‘( bday 𝐴)) → ( bday 𝑥L) ∈ ( bday 𝐴))
6765, 66syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → ( bday 𝑥L) ∈ ( bday 𝐴))
6810a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ( R ‘𝐴) ⊆ ( O ‘( bday 𝐴)))
6968sselda 3942 . . . . . . . . . . . . 13 ((𝜑𝑥R ∈ ( R ‘𝐴)) → 𝑥R ∈ ( O ‘( bday 𝐴)))
7069adantrr 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → 𝑥R ∈ ( O ‘( bday 𝐴)))
71 oldbdayim 27168 . . . . . . . . . . . 12 (𝑥R ∈ ( O ‘( bday 𝐴)) → ( bday 𝑥R) ∈ ( bday 𝐴))
7270, 71syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → ( bday 𝑥R) ∈ ( bday 𝐴))
73 bdayelon 27067 . . . . . . . . . . . 12 ( bday 𝑥L) ∈ On
74 bdayelon 27067 . . . . . . . . . . . 12 ( bday 𝑥R) ∈ On
75 bdayelon 27067 . . . . . . . . . . . 12 ( bday 𝐴) ∈ On
76 onunel 34103 . . . . . . . . . . . 12 ((( bday 𝑥L) ∈ On ∧ ( bday 𝑥R) ∈ On ∧ ( bday 𝐴) ∈ On) → ((( bday 𝑥L) ∪ ( bday 𝑥R)) ∈ ( bday 𝐴) ↔ (( bday 𝑥L) ∈ ( bday 𝐴) ∧ ( bday 𝑥R) ∈ ( bday 𝐴))))
7773, 74, 75, 76mp3an 1461 . . . . . . . . . . 11 ((( bday 𝑥L) ∪ ( bday 𝑥R)) ∈ ( bday 𝐴) ↔ (( bday 𝑥L) ∈ ( bday 𝐴) ∧ ( bday 𝑥R) ∈ ( bday 𝐴)))
7867, 72, 77sylanbrc 583 . . . . . . . . . 10 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → (( bday 𝑥L) ∪ ( bday 𝑥R)) ∈ ( bday 𝐴))
79 elun1 4134 . . . . . . . . . 10 ((( bday 𝑥L) ∪ ( bday 𝑥R)) ∈ ( bday 𝐴) → (( bday 𝑥L) ∪ ( bday 𝑥R)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
8078, 79syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → (( bday 𝑥L) ∪ ( bday 𝑥R)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
8158, 60, 63, 80negsproplem1 34314 . . . . . . . 8 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → (( -us ‘𝑥L) ∈ No ∧ (𝑥L <s 𝑥R → ( -us ‘𝑥R) <s ( -us ‘𝑥L))))
8281simprd 496 . . . . . . 7 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → (𝑥L <s 𝑥R → ( -us ‘𝑥R) <s ( -us ‘𝑥L)))
8357, 82mpd 15 . . . . . 6 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → ( -us ‘𝑥R) <s ( -us ‘𝑥L))
84 breq12 5108 . . . . . 6 ((( -us ‘𝑥R) = 𝑎 ∧ ( -us ‘𝑥L) = 𝑏) → (( -us ‘𝑥R) <s ( -us ‘𝑥L) ↔ 𝑎 <s 𝑏))
8583, 84syl5ibcom 244 . . . . 5 ((𝜑 ∧ (𝑥R ∈ ( R ‘𝐴) ∧ 𝑥L ∈ ( L ‘𝐴))) → ((( -us ‘𝑥R) = 𝑎 ∧ ( -us ‘𝑥L) = 𝑏) → 𝑎 <s 𝑏))
8685rexlimdvva 3203 . . . 4 (𝜑 → (∃𝑥R ∈ ( R ‘𝐴)∃𝑥L ∈ ( L ‘𝐴)(( -us ‘𝑥R) = 𝑎 ∧ ( -us ‘𝑥L) = 𝑏) → 𝑎 <s 𝑏))
8750, 86biimtrid 241 . . 3 (𝜑 → ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏))
88873impib 1116 . 2 ((𝜑𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏)
896, 9, 37, 41, 88ssltd 27082 1 (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  wrex 3071  Vcvv 3443  cun 3906  wss 3908  c0 4280   class class class wbr 5103  dom cdm 5631  cima 5634  Oncon0 6315  Fun wfun 6487   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:  negsproplem3  34316
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