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Theorem negsproplem2 28062
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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
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 28056 . . . 4 -us Fn No
2 fnfun 6667 . . . 4 ( -us Fn No → Fun -us )
31, 2ax-mp 5 . . 3 Fun -us
4 fvex 6918 . . . 4 ( R ‘𝐴) ∈ V
54funimaex 6654 . . 3 (Fun -us → ( -us “ ( R ‘𝐴)) ∈ V)
63, 5mp1i 13 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) ∈ V)
7 fvex 6918 . . . 4 ( L ‘𝐴) ∈ V
87funimaex 6654 . . 3 (Fun -us → ( -us “ ( L ‘𝐴)) ∈ V)
93, 8mp1i 13 . 2 (𝜑 → ( -us “ ( L ‘𝐴)) ∈ V)
10 rightssold 27919 . . . 4 ( R ‘𝐴) ⊆ ( O ‘( bday 𝐴))
11 imass2 6119 . . . 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)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
1413adantr 480 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
15 oldssno 27901 . . . . . . . . 9 ( O ‘( bday 𝐴)) ⊆ No
1615sseli 3978 . . . . . . . 8 (𝑎 ∈ ( O ‘( bday 𝐴)) → 𝑎 No )
1716adantl 481 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → 𝑎 No )
18 0sno 27872 . . . . . . . 8 0s No
1918a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → 0s No )
20 bday0s 27874 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
2120uneq2i 4164 . . . . . . . . 9 (( bday 𝑎) ∪ ( bday ‘ 0s )) = (( bday 𝑎) ∪ ∅)
22 un0 4393 . . . . . . . . 9 (( bday 𝑎) ∪ ∅) = ( bday 𝑎)
2321, 22eqtri 2764 . . . . . . . 8 (( bday 𝑎) ∪ ( bday ‘ 0s )) = ( bday 𝑎)
24 oldbdayim 27928 . . . . . . . . . 10 (𝑎 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑎) ∈ ( bday 𝐴))
2524adantl 481 . . . . . . . . 9 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ( bday 𝑎) ∈ ( bday 𝐴))
26 elun1 4181 . . . . . . . . 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 2844 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → (( bday 𝑎) ∪ ( bday ‘ 0s )) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2914, 17, 19, 28negsproplem1 28061 . . . . . 6 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → (( -us𝑎) ∈ No ∧ (𝑎 <s 0s → ( -us ‘ 0s ) <s ( -us𝑎))))
3029simpld 494 . . . . 5 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ( -us𝑎) ∈ No )
3130ralrimiva 3145 . . . 4 (𝜑 → ∀𝑎 ∈ ( O ‘( bday 𝐴))( -us𝑎) ∈ No )
321fndmi 6671 . . . . . 6 dom -us = No
3315, 32sseqtrri 4032 . . . . 5 ( O ‘( bday 𝐴)) ⊆ dom -us
34 funimass4 6972 . . . . 5 ((Fun -us ∧ ( O ‘( bday 𝐴)) ⊆ dom -us ) → (( -us “ ( O ‘( bday 𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday 𝐴))( -us𝑎) ∈ No ))
353, 33, 34mp2an 692 . . . 4 (( -us “ ( O ‘( bday 𝐴))) ⊆ No ↔ ∀𝑎 ∈ ( O ‘( bday 𝐴))( -us𝑎) ∈ No )
3631, 35sylibr 234 . . 3 (𝜑 → ( -us “ ( O ‘( bday 𝐴))) ⊆ No )
3712, 36sstrid 3994 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) ⊆ No )
38 leftssold 27918 . . . 4 ( L ‘𝐴) ⊆ ( O ‘( bday 𝐴))
39 imass2 6119 . . . 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 3994 . 2 (𝜑 → ( -us “ ( L ‘𝐴)) ⊆ No )
42 rightssno 27921 . . . . . . 7 ( R ‘𝐴) ⊆ No
43 fvelimab 6980 . . . . . . 7 (( -us Fn No ∧ ( R ‘𝐴) ⊆ No ) → (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us𝑥𝑅) = 𝑎))
441, 42, 43mp2an 692 . . . . . 6 (𝑎 ∈ ( -us “ ( R ‘𝐴)) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)( -us𝑥𝑅) = 𝑎)
45 leftssno 27920 . . . . . . 7 ( L ‘𝐴) ⊆ No
46 fvelimab 6980 . . . . . . 7 (( -us Fn No ∧ ( L ‘𝐴) ⊆ No ) → (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us𝑥𝐿) = 𝑏))
471, 45, 46mp2an 692 . . . . . 6 (𝑏 ∈ ( -us “ ( L ‘𝐴)) ↔ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us𝑥𝐿) = 𝑏)
4844, 47anbi12i 628 . . . . 5 ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)( -us𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us𝑥𝐿) = 𝑏))
49 reeanv 3228 . . . . 5 (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us𝑥𝑅) = 𝑎 ∧ ( -us𝑥𝐿) = 𝑏) ↔ (∃𝑥𝑅 ∈ ( R ‘𝐴)( -us𝑥𝑅) = 𝑎 ∧ ∃𝑥𝐿 ∈ ( L ‘𝐴)( -us𝑥𝐿) = 𝑏))
5048, 49bitr4i 278 . . . 4 ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) ↔ ∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us𝑥𝑅) = 𝑎 ∧ ( -us𝑥𝐿) = 𝑏))
51 lltropt 27912 . . . . . . . . 9 ( L ‘𝐴) <<s ( R ‘𝐴)
5251a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( L ‘𝐴) <<s ( R ‘𝐴))
53 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( L ‘𝐴))
54 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( R ‘𝐴))
5552, 53, 54ssltsepcd 27840 . . . . . . 7 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 <s 𝑥𝑅)
5613adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ∀𝑥 No 𝑦 No ((( bday 𝑥) ∪ ( bday 𝑦)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)) → (( -us𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -us𝑦) <s ( -us𝑥)))))
5745sseli 3978 . . . . . . . . . 10 (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 No )
5857ad2antll 729 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 No )
5942sseli 3978 . . . . . . . . . . 11 (𝑥𝑅 ∈ ( R ‘𝐴) → 𝑥𝑅 No )
6059adantr 480 . . . . . . . . . 10 ((𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴)) → 𝑥𝑅 No )
6160adantl 481 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 No )
6238sseli 3978 . . . . . . . . . . . . 13 (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ ( O ‘( bday 𝐴)))
6362ad2antll 729 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( O ‘( bday 𝐴)))
64 oldbdayim 27928 . . . . . . . . . . . 12 (𝑥𝐿 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑥𝐿) ∈ ( bday 𝐴))
6563, 64syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( bday 𝑥𝐿) ∈ ( bday 𝐴))
6610a1i 11 . . . . . . . . . . . . . 14 (𝜑 → ( R ‘𝐴) ⊆ ( O ‘( bday 𝐴)))
6766sselda 3982 . . . . . . . . . . . . 13 ((𝜑𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ ( O ‘( bday 𝐴)))
6867adantrr 717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( O ‘( bday 𝐴)))
69 oldbdayim 27928 . . . . . . . . . . . 12 (𝑥𝑅 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑥𝑅) ∈ ( bday 𝐴))
7068, 69syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( bday 𝑥𝑅) ∈ ( bday 𝐴))
71 bdayelon 27822 . . . . . . . . . . . 12 ( bday 𝑥𝐿) ∈ On
72 bdayelon 27822 . . . . . . . . . . . 12 ( bday 𝑥𝑅) ∈ On
73 bdayelon 27822 . . . . . . . . . . . 12 ( bday 𝐴) ∈ On
74 onunel 6488 . . . . . . . . . . . 12 ((( bday 𝑥𝐿) ∈ On ∧ ( bday 𝑥𝑅) ∈ On ∧ ( bday 𝐴) ∈ On) → ((( bday 𝑥𝐿) ∪ ( bday 𝑥𝑅)) ∈ ( bday 𝐴) ↔ (( bday 𝑥𝐿) ∈ ( bday 𝐴) ∧ ( bday 𝑥𝑅) ∈ ( bday 𝐴))))
7571, 72, 73, 74mp3an 1462 . . . . . . . . . . 11 ((( bday 𝑥𝐿) ∪ ( bday 𝑥𝑅)) ∈ ( bday 𝐴) ↔ (( bday 𝑥𝐿) ∈ ( bday 𝐴) ∧ ( bday 𝑥𝑅) ∈ ( bday 𝐴)))
7665, 70, 75sylanbrc 583 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday 𝑥𝐿) ∪ ( bday 𝑥𝑅)) ∈ ( bday 𝐴))
77 elun1 4181 . . . . . . . . . 10 ((( bday 𝑥𝐿) ∪ ( bday 𝑥𝑅)) ∈ ( bday 𝐴) → (( bday 𝑥𝐿) ∪ ( bday 𝑥𝑅)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
7876, 77syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( bday 𝑥𝐿) ∪ ( bday 𝑥𝑅)) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
7956, 58, 61, 78negsproplem1 28061 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (( -us𝑥𝐿) ∈ No ∧ (𝑥𝐿 <s 𝑥𝑅 → ( -us𝑥𝑅) <s ( -us𝑥𝐿))))
8079simprd 495 . . . . . . 7 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → (𝑥𝐿 <s 𝑥𝑅 → ( -us𝑥𝑅) <s ( -us𝑥𝐿)))
8155, 80mpd 15 . . . . . 6 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( -us𝑥𝑅) <s ( -us𝑥𝐿))
82 breq12 5147 . . . . . 6 ((( -us𝑥𝑅) = 𝑎 ∧ ( -us𝑥𝐿) = 𝑏) → (( -us𝑥𝑅) <s ( -us𝑥𝐿) ↔ 𝑎 <s 𝑏))
8381, 82syl5ibcom 245 . . . . 5 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ((( -us𝑥𝑅) = 𝑎 ∧ ( -us𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
8483rexlimdvva 3212 . . . 4 (𝜑 → (∃𝑥𝑅 ∈ ( R ‘𝐴)∃𝑥𝐿 ∈ ( L ‘𝐴)(( -us𝑥𝑅) = 𝑎 ∧ ( -us𝑥𝐿) = 𝑏) → 𝑎 <s 𝑏))
8550, 84biimtrid 242 . . 3 (𝜑 → ((𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏))
86853impib 1116 . 2 ((𝜑𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏)
876, 9, 37, 41, 86ssltd 27837 1 (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cun 3948  wss 3950  c0 4332   class class class wbr 5142  dom cdm 5684  cima 5687  Oncon0 6383  Fun wfun 6554   Fn wfn 6555  cfv 6560   No csur 27685   <s cslt 27686   bday cbday 27687   <<s csslt 27826   0s c0s 27868   O cold 27883   L cleft 27885   R cright 27886   -us cnegs 28052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-no 27688  df-slt 27689  df-bday 27690  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-negs 28054
This theorem is referenced by:  negsproplem3  28063
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