Theorem ltsrec 28074

Description: A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)

Assertion

Ref	Expression
ltsrec	⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))

Distinct variable groups:   𝐴,𝑏,𝑐   𝐵,𝑏,𝑐   𝐶,𝑏,𝑐   𝐷,𝑏,𝑐   𝑋,𝑏,𝑐   𝑌,𝑏,𝑐

Proof of Theorem ltsrec

Step	Hyp	Ref	Expression
1		simplr 781	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 <<s 𝐷)
2		simpll 779	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐴 <<s 𝐵)
3		simprr 785	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 = (𝐶 |s 𝐷))
4		simprl 783	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 = (𝐴 |s 𝐵))
5	1, 2, 3, 4	lesrecd 28073	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ (∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ∧ ∀𝑐 ∈ 𝐶 𝑐 <s 𝑋)))
6		ancom 466	. . . 4 ⊢ ((∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ∧ ∀𝑐 ∈ 𝐶 𝑐 <s 𝑋) ↔ (∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏))
7	5, 6	bitrdi 290	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ (∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏)))
8		cutcuts 28054	. . . . . . 7 ⊢ (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
9	8	simp1d 1160	. . . . . 6 ⊢ (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
10	9	ad2antlr 740	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
11	3, 10	eqeltrd 2862	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 ∈ No )
12		cutcuts 28054	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
13	12	simp1d 1160	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
14	13	ad2antrr 739	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
15	4, 14	eqeltrd 2862	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 ∈ No )
16		lenlts 27996	. . . 4 ⊢ ((𝑌 ∈ No ∧ 𝑋 ∈ No ) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
17	11, 15, 16	syl2anc 596	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
18		sltsss1 28038	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
19	18	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 ⊆ No )
20	19	sselda 3934	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ No )
21	15	adantr 486	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑋 ∈ No )
22		ltnles 27997	. . . . . . 7 ⊢ ((𝑐 ∈ No ∧ 𝑋 ∈ No ) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
23	20, 21, 22	syl2anc 596	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
24	23	ralbidva 3185	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ↔ ∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐))
25	11	adantr 486	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑌 ∈ No )
26		sltsss2 28039	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
27	26	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐵 ⊆ No )
28	27	sselda 3934	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ No )
29		ltnles 27997	. . . . . . 7 ⊢ ((𝑌 ∈ No ∧ 𝑏 ∈ No ) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
30	25, 28, 29	syl2anc 596	. . . . . 6 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
31	30	ralbidva 3185	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (∀𝑏 ∈ 𝐵 𝑌 <s 𝑏 ↔ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌))
32	24, 31	anbi12d 644	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → ((∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏) ↔ (∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌)))
33		ralnex 3090	. . . . . 6 ⊢ (∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ↔ ¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐)
34		ralnex 3090	. . . . . 6 ⊢ (∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌 ↔ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)
35	33, 34	anbi12i 640	. . . . 5 ⊢ ((∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
36		ioran 999	. . . . 5 ⊢ (¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌) ↔ (¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∧ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
37	35, 36	bitr4i 281	. . . 4 ⊢ ((∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ∧ ∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌) ↔ ¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌))
38	32, 37	bitrdi 290	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → ((∀𝑐 ∈ 𝐶 𝑐 <s 𝑋 ∧ ∀𝑏 ∈ 𝐵 𝑌 <s 𝑏) ↔ ¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
39	7, 17, 38	3bitr3d 312	. 2 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (¬ 𝑋 <s 𝑌 ↔ ¬ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))
40	39	con4bid 320	1 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑋 <s 𝑌 ↔ (∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐 ∨ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902  {csn 4587   class class class wbr 5107  (class class class)co 7417   No csur 27884   <s clts 27885   ≤s cles 27988   <<s cslts 28030   |s ccuts 28032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1o 8459  df-2o 8460  df-no 27887  df-lts 27888  df-bday 27889  df-les 27989  df-slts 28031  df-cuts 28033

This theorem is used by:  ltsrecd  28075  0lt1s  28085