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Theorem negsproplem2 28076
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 28070 . . . 4 -us Fn No
2 fnfun 6669 . . . 4 ( -us Fn No → Fun -us )
31, 2ax-mp 5 . . 3 Fun -us
4 fvex 6920 . . . 4 ( R ‘𝐴) ∈ V
54funimaex 6656 . . 3 (Fun -us → ( -us “ ( R ‘𝐴)) ∈ V)
63, 5mp1i 13 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) ∈ V)
7 fvex 6920 . . . 4 ( L ‘𝐴) ∈ V
87funimaex 6656 . . 3 (Fun -us → ( -us “ ( L ‘𝐴)) ∈ V)
93, 8mp1i 13 . 2 (𝜑 → ( -us “ ( L ‘𝐴)) ∈ V)
10 rightssold 27933 . . . 4 ( R ‘𝐴) ⊆ ( O ‘( bday 𝐴))
11 imass2 6123 . . . 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 27915 . . . . . . . . 9 ( O ‘( bday 𝐴)) ⊆ No
1615sseli 3991 . . . . . . . 8 (𝑎 ∈ ( O ‘( bday 𝐴)) → 𝑎 No )
1716adantl 481 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → 𝑎 No )
18 0sno 27886 . . . . . . . 8 0s No
1918a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → 0s No )
20 bday0s 27888 . . . . . . . . . 10 ( bday ‘ 0s ) = ∅
2120uneq2i 4175 . . . . . . . . 9 (( bday 𝑎) ∪ ( bday ‘ 0s )) = (( bday 𝑎) ∪ ∅)
22 un0 4400 . . . . . . . . 9 (( bday 𝑎) ∪ ∅) = ( bday 𝑎)
2321, 22eqtri 2763 . . . . . . . 8 (( bday 𝑎) ∪ ( bday ‘ 0s )) = ( bday 𝑎)
24 oldbdayim 27942 . . . . . . . . . 10 (𝑎 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑎) ∈ ( bday 𝐴))
2524adantl 481 . . . . . . . . 9 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ( bday 𝑎) ∈ ( bday 𝐴))
26 elun1 4192 . . . . . . . . 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 2843 . . . . . . 7 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → (( bday 𝑎) ∪ ( bday ‘ 0s )) ∈ (( bday 𝐴) ∪ ( bday 𝐵)))
2914, 17, 19, 28negsproplem1 28075 . . . . . 6 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → (( -us𝑎) ∈ No ∧ (𝑎 <s 0s → ( -us ‘ 0s ) <s ( -us𝑎))))
3029simpld 494 . . . . 5 ((𝜑𝑎 ∈ ( O ‘( bday 𝐴))) → ( -us𝑎) ∈ No )
3130ralrimiva 3144 . . . 4 (𝜑 → ∀𝑎 ∈ ( O ‘( bday 𝐴))( -us𝑎) ∈ No )
321fndmi 6673 . . . . . 6 dom -us = No
3315, 32sseqtrri 4033 . . . . 5 ( O ‘( bday 𝐴)) ⊆ dom -us
34 funimass4 6973 . . . . 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 4007 . 2 (𝜑 → ( -us “ ( R ‘𝐴)) ⊆ No )
38 leftssold 27932 . . . 4 ( L ‘𝐴) ⊆ ( O ‘( bday 𝐴))
39 imass2 6123 . . . 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 4007 . 2 (𝜑 → ( -us “ ( L ‘𝐴)) ⊆ No )
42 rightssno 27935 . . . . . . 7 ( R ‘𝐴) ⊆ No
43 fvelimab 6981 . . . . . . 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 27934 . . . . . . 7 ( L ‘𝐴) ⊆ No
46 fvelimab 6981 . . . . . . 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 3227 . . . . 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 27926 . . . . . . . . 9 ( L ‘𝐴) <<s ( R ‘𝐴)
5251a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → ( L ‘𝐴) <<s ( R ‘𝐴))
53 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( L ‘𝐴))
54 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( R ‘𝐴))
5552, 53, 54ssltsepcd 27854 . . . . . . 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 3991 . . . . . . . . . 10 (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 No )
5857ad2antll 729 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 No )
5942sseli 3991 . . . . . . . . . . 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 3991 . . . . . . . . . . . . 13 (𝑥𝐿 ∈ ( L ‘𝐴) → 𝑥𝐿 ∈ ( O ‘( bday 𝐴)))
6362ad2antll 729 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝐿 ∈ ( O ‘( bday 𝐴)))
64 oldbdayim 27942 . . . . . . . . . . . 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 3995 . . . . . . . . . . . . 13 ((𝜑𝑥𝑅 ∈ ( R ‘𝐴)) → 𝑥𝑅 ∈ ( O ‘( bday 𝐴)))
6867adantrr 717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝑅 ∈ ( R ‘𝐴) ∧ 𝑥𝐿 ∈ ( L ‘𝐴))) → 𝑥𝑅 ∈ ( O ‘( bday 𝐴)))
69 oldbdayim 27942 . . . . . . . . . . . 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 27836 . . . . . . . . . . . 12 ( bday 𝑥𝐿) ∈ On
72 bdayelon 27836 . . . . . . . . . . . 12 ( bday 𝑥𝑅) ∈ On
73 bdayelon 27836 . . . . . . . . . . . 12 ( bday 𝐴) ∈ On
74 onunel 6491 . . . . . . . . . . . 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 1460 . . . . . . . . . . 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 4192 . . . . . . . . . 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 28075 . . . . . . . 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 5153 . . . . . 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 3211 . . . 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 1115 . 2 ((𝜑𝑎 ∈ ( -us “ ( R ‘𝐴)) ∧ 𝑏 ∈ ( -us “ ( L ‘𝐴))) → 𝑎 <s 𝑏)
876, 9, 37, 41, 86ssltd 27851 1 (𝜑 → ( -us “ ( R ‘𝐴)) <<s ( -us “ ( L ‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  cun 3961  wss 3963  c0 4339   class class class wbr 5148  dom cdm 5689  cima 5692  Oncon0 6386  Fun wfun 6557   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:  negsproplem3  28077
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