Theorem ltsrec 27952

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 780	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 <<s 𝐷)
2		simpll 778	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐴 <<s 𝐵)
3		simprr 784	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 = (𝐶 |s 𝐷))
4		simprl 782	. . . . 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 27951	. . . 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 465	. . . 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 27932	. . . . . . 7 ⊢ (𝐶 <<s 𝐷 → ((𝐶 |s 𝐷) ∈ No ∧ 𝐶 <<s {(𝐶 |s 𝐷)} ∧ {(𝐶 |s 𝐷)} <<s 𝐷))
9	8	simp1d 1158	. . . . . 6 ⊢ (𝐶 <<s 𝐷 → (𝐶 |s 𝐷) ∈ No )
10	9	ad2antlr 739	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐶 |s 𝐷) ∈ No )
11	3, 10	eqeltrd 2865	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑌 ∈ No )
12		cutcuts 27932	. . . . . . 7 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
13	12	simp1d 1158	. . . . . 6 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ No )
14	13	ad2antrr 738	. . . . 5 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝐴 |s 𝐵) ∈ No )
15	4, 14	eqeltrd 2865	. . . 4 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝑋 ∈ No )
16		lenlts 27874	. . . 4 ⊢ ((𝑌 ∈ No ∧ 𝑋 ∈ No ) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
17	11, 15, 16	syl2anc 595	. . 3 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → (𝑌 ≤s 𝑋 ↔ ¬ 𝑋 <s 𝑌))
18		sltsss1 27916	. . . . . . . . 9 ⊢ (𝐶 <<s 𝐷 → 𝐶 ⊆ No )
19	18	ad2antlr 739	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐶 ⊆ No )
20	19	sselda 3939	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ No )
21	15	adantr 485	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑐 ∈ 𝐶) → 𝑋 ∈ No )
22		ltnles 27875	. . . . . . 7 ⊢ ((𝑐 ∈ No ∧ 𝑋 ∈ No ) → (𝑐 <s 𝑋 ↔ ¬ 𝑋 ≤s 𝑐))
23	20, 21, 22	syl2anc 595	. . . . . 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 3186	. . . . 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 485	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑌 ∈ No )
26		sltsss2 27917	. . . . . . . . 9 ⊢ (𝐴 <<s 𝐵 → 𝐵 ⊆ No )
27	26	ad2antrr 738	. . . . . . . 8 ⊢ (((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) → 𝐵 ⊆ No )
28	27	sselda 3939	. . . . . . 7 ⊢ ((((𝐴 <<s 𝐵 ∧ 𝐶 <<s 𝐷) ∧ (𝑋 = (𝐴 |s 𝐵) ∧ 𝑌 = (𝐶 |s 𝐷))) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ No )
29		ltnles 27875	. . . . . . 7 ⊢ ((𝑌 ∈ No ∧ 𝑏 ∈ No ) → (𝑌 <s 𝑏 ↔ ¬ 𝑏 ≤s 𝑌))
30	25, 28, 29	syl2anc 595	. . . . . 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 3186	. . . . 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 643	. . . 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 3091	. . . . . 6 ⊢ (∀𝑐 ∈ 𝐶 ¬ 𝑋 ≤s 𝑐 ↔ ¬ ∃𝑐 ∈ 𝐶 𝑋 ≤s 𝑐)
34		ralnex 3091	. . . . . 6 ⊢ (∀𝑏 ∈ 𝐵 ¬ 𝑏 ≤s 𝑌 ↔ ¬ ∃𝑏 ∈ 𝐵 𝑏 ≤s 𝑌)
35	33, 34	anbi12i 639	. . . . 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)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ⊆ wss 3907  {csn 4585   class class class wbr 5105  (class class class)co 7400   No csur 27762   <s clts 27763   ≤s cles 27866   <<s cslts 27908   |s ccuts 27910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1o 8441  df-2o 8442  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911

This theorem is referenced by:  ltsrecd  27953  0lt1s  27963