Theorem supsrlem 11022

Description: Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
supsrlem.1	⊢ 𝐵 = {𝑤 ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}
supsrlem.2	⊢ 𝐶 ∈ R

Assertion

Ref	Expression
supsrlem	⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤   𝑥,𝐶,𝑦,𝑧,𝑤

Proof of Theorem supsrlem

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supsrlem.2	. . . . . . 7 ⊢ 𝐶 ∈ R
2		0idsr 11008	. . . . . . 7 ⊢ (𝐶 ∈ R → (𝐶 +R 0R) = 𝐶)
3	1, 2	mp1i 13	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → (𝐶 +R 0R) = 𝐶)
4		simpl 482	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 𝐶 ∈ 𝐴)
5	3, 4	eqeltrd 2836	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → (𝐶 +R 0R) ∈ 𝐴)
6		1pr 10926	. . . . . . 7 ⊢ 1P ∈ P
7	6	elexi 3463	. . . . . 6 ⊢ 1P ∈ V
8		opeq1 4829	. . . . . . . . . 10 ⊢ (𝑤 = 1P → ⟨𝑤, 1P⟩ = ⟨1P, 1P⟩)
9	8	eceq1d 8675	. . . . . . . . 9 ⊢ (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = [⟨1P, 1P⟩] ~R )
10		df-0r 10971	. . . . . . . . 9 ⊢ 0R = [⟨1P, 1P⟩] ~R
11	9, 10	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = 0R)
12	11	oveq2d 7374	. . . . . . 7 ⊢ (𝑤 = 1P → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R 0R))
13	12	eleq1d 2821	. . . . . 6 ⊢ (𝑤 = 1P → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ (𝐶 +R 0R) ∈ 𝐴))
14		supsrlem.1	. . . . . 6 ⊢ 𝐵 = {𝑤 ∣ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴}
15	7, 13, 14	elab2 3637	. . . . 5 ⊢ (1P ∈ 𝐵 ↔ (𝐶 +R 0R) ∈ 𝐴)
16	5, 15	sylibr 234	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 1P ∈ 𝐵)
17	16	ne0d 4294	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 𝐵 ≠ ∅)
18		breq1 5101	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦 <R 𝑥 ↔ 𝐶 <R 𝑥))
19	18	rspccv 3573	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → 𝐶 <R 𝑥))
20		0lt1sr 11006	. . . . . . . . . . . . 13 ⊢ 0R <R 1R
21		m1r 10993	. . . . . . . . . . . . . 14 ⊢ -1R ∈ R
22		ltasr 11011	. . . . . . . . . . . . . 14 ⊢ (-1R ∈ R → (0R <R 1R ↔ (-1R +R 0R) <R (-1R +R 1R)))
23	21, 22	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (0R <R 1R ↔ (-1R +R 0R) <R (-1R +R 1R))
24	20, 23	mpbi 230	. . . . . . . . . . . 12 ⊢ (-1R +R 0R) <R (-1R +R 1R)
25		0idsr 11008	. . . . . . . . . . . . 13 ⊢ (-1R ∈ R → (-1R +R 0R) = -1R)
26	21, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ (-1R +R 0R) = -1R
27		m1p1sr 11003	. . . . . . . . . . . 12 ⊢ (-1R +R 1R) = 0R
28	24, 26, 27	3brtr3i 5127	. . . . . . . . . . 11 ⊢ -1R <R 0R
29		ltasr 11011	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ R → (-1R <R 0R ↔ (𝐶 +R -1R) <R (𝐶 +R 0R)))
30	1, 29	ax-mp 5	. . . . . . . . . . 11 ⊢ (-1R <R 0R ↔ (𝐶 +R -1R) <R (𝐶 +R 0R))
31	28, 30	mpbi 230	. . . . . . . . . 10 ⊢ (𝐶 +R -1R) <R (𝐶 +R 0R)
32	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐶 +R 0R) = 𝐶
33	31, 32	breqtri 5123	. . . . . . . . 9 ⊢ (𝐶 +R -1R) <R 𝐶
34		ltsosr 11005	. . . . . . . . . 10 ⊢ <R Or R
35		ltrelsr 10979	. . . . . . . . . 10 ⊢ <R ⊆ (R × R)
36	34, 35	sotri 6084	. . . . . . . . 9 ⊢ (((𝐶 +R -1R) <R 𝐶 ∧ 𝐶 <R 𝑥) → (𝐶 +R -1R) <R 𝑥)
37	33, 36	mpan 690	. . . . . . . 8 ⊢ (𝐶 <R 𝑥 → (𝐶 +R -1R) <R 𝑥)
38	1	map2psrpr 11021	. . . . . . . 8 ⊢ ((𝐶 +R -1R) <R 𝑥 ↔ ∃𝑣 ∈ P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥)
39	37, 38	sylib 218	. . . . . . 7 ⊢ (𝐶 <R 𝑥 → ∃𝑣 ∈ P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥)
40	19, 39	syl6 35	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → ∃𝑣 ∈ P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥))
41		breq2 5102	. . . . . . . . . 10 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑦 <R 𝑥))
42	41	ralbidv 3159	. . . . . . . . 9 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥))
43	14	eqabri 2878	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴)
44		breq1 5101	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝐶 +R [⟨𝑤, 1P⟩] ~R ) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
45	44	rspccv 3573	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
46	1	ltpsrpr 11020	. . . . . . . . . . . 12 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑤<P 𝑣)
47	45, 46	imbitrdi 251	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 → 𝑤<P 𝑣))
48	43, 47	biimtrid 242	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑤 ∈ 𝐵 → 𝑤<P 𝑣))
49	48	ralrimiv 3127	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∀𝑤 ∈ 𝐵 𝑤<P 𝑣)
50	42, 49	biimtrrdi 254	. . . . . . . 8 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → ∀𝑤 ∈ 𝐵 𝑤<P 𝑣))
51	50	com12 32	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → ∀𝑤 ∈ 𝐵 𝑤<P 𝑣))
52	51	reximdv 3151	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (∃𝑣 ∈ P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣))
53	40, 52	syld 47	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣))
54	53	rexlimivw 3133	. . . 4 ⊢ (∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣))
55	54	impcom 407	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣)
56		supexpr 10965	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣) → ∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)))
57	17, 55, 56	syl2anc 584	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)))
58	1	mappsrpr 11019	. . . . . . 7 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑣 ∈ P)
59	35	brel 5689	. . . . . . 7 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
60	58, 59	sylbir 235	. . . . . 6 ⊢ (𝑣 ∈ P → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
61	60	simprd 495	. . . . 5 ⊢ (𝑣 ∈ P → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
62	61	adantl 481	. . . 4 ⊢ (((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
63	34, 35	sotri 6084	. . . . . . . . . . . . . . 15 ⊢ (((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (𝐶 +R -1R) <R 𝑦)
64	58, 63	sylanbr 582	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (𝐶 +R -1R) <R 𝑦)
65	1	map2psrpr 11021	. . . . . . . . . . . . . 14 ⊢ ((𝐶 +R -1R) <R 𝑦 ↔ ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
66	64, 65	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
67		rexex 3066	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
68		df-ral 3052	. . . . . . . . . . . . . . 15 ⊢ (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ↔ ∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤))
69		19.29 1874	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ∧ ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → ∃𝑤((𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦))
70		eleq1 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
71	43, 70	bitrid 283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤 ∈ 𝐵 ↔ 𝑦 ∈ 𝐴))
72	1	ltpsrpr 11020	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ↔ 𝑣<P 𝑤)
73		breq2 5102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
74	72, 73	bitr3id 285	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑣<P 𝑤 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
75	74	notbid 318	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (¬ 𝑣<P 𝑤 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
76	71, 75	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ↔ (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
77	76	biimpac 478	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
78	77	exlimiv 1931	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑤((𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
79	69, 78	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑤(𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ∧ ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
80	68, 79	sylanb 581	. . . . . . . . . . . . . 14 ⊢ ((∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
81	80	expcom 413	. . . . . . . . . . . . 13 ⊢ (∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
82	66, 67, 81	3syl 18	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
83	82	impd 410	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → ((∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ 𝑦 ∈ 𝐴) → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
84	83	impancom 451	. . . . . . . . . 10 ⊢ ((𝑣 ∈ P ∧ (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ 𝑦 ∈ 𝐴)) → ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
85	84	pm2.01d 190	. . . . . . . . 9 ⊢ ((𝑣 ∈ P ∧ (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
86	85	expr 456	. . . . . . . 8 ⊢ ((𝑣 ∈ P ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤) → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
87	86	ralrimiv 3127	. . . . . . 7 ⊢ ((𝑣 ∈ P ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤) → ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
88	87	ex 412	. . . . . 6 ⊢ (𝑣 ∈ P → (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 → ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
89	88	adantl 481	. . . . 5 ⊢ (((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 → ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
90		r19.29 3099	. . . . . . . . . . . . . 14 ⊢ ((∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) ∧ ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → ∃𝑤 ∈ P ((𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦))
91		breq1 5101	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
92	46, 91	bitr3id 285	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤<P 𝑣 ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
93	92	biimprd 248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → 𝑤<P 𝑣))
94		vex 3444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑢 ∈ V
95		opeq1 4829	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑢 → ⟨𝑤, 1P⟩ = ⟨𝑢, 1P⟩)
96	95	eceq1d 8675	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑢 → [⟨𝑤, 1P⟩] ~R = [⟨𝑢, 1P⟩] ~R )
97	96	oveq2d 7374	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑢 → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R [⟨𝑢, 1P⟩] ~R ))
98	97	eleq1d 2821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑢 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴))
99	94, 98, 14	elab2 3637	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ 𝐵 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴)
100		breq2 5102	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = (𝐶 +R [⟨𝑢, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑢, 1P⟩] ~R )))
101	1	ltpsrpr 11020	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ↔ 𝑤<P 𝑢)
102	100, 101	bitrdi 287	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝐶 +R [⟨𝑢, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ 𝑤<P 𝑢))
103	102	rspcev 3576	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴 ∧ 𝑤<P 𝑢) → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
104	99, 103	sylanb 581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤<P 𝑢) → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
105	104	rexlimiva 3129	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑢 ∈ 𝐵 𝑤<P 𝑢 → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
106		breq1 5101	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ 𝑦 <R 𝑧))
107	106	rexbidv 3160	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
108	105, 107	imbitrid 244	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (∃𝑢 ∈ 𝐵 𝑤<P 𝑢 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
109	93, 108	imim12d 81	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
110	109	impcom 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
111	110	rexlimivw 3133	. . . . . . . . . . . . . 14 ⊢ (∃𝑤 ∈ P ((𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) ∧ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
112	90, 111	syl 17	. . . . . . . . . . . . 13 ⊢ ((∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) ∧ ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
113	65, 112	sylan2b 594	. . . . . . . . . . . 12 ⊢ ((∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) ∧ (𝐶 +R -1R) <R 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
114	113	ex 412	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) → ((𝐶 +R -1R) <R 𝑦 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
115	114	adantl 481	. . . . . . . . . 10 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝑣 ∈ P) ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → ((𝐶 +R -1R) <R 𝑦 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
116	115	a1dd 50	. . . . . . . . 9 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝑣 ∈ P) ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → ((𝐶 +R -1R) <R 𝑦 → (𝑦 ∈ R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))))
117	34, 35	sotri2 6086	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ ¬ (𝐶 +R -1R) <R 𝑦 ∧ (𝐶 +R -1R) <R 𝐶) → 𝑦 <R 𝐶)
118	33, 117	mp3an3 1452	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ ¬ (𝐶 +R -1R) <R 𝑦) → 𝑦 <R 𝐶)
119		breq2 5102	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐶 → (𝑦 <R 𝑧 ↔ 𝑦 <R 𝐶))
120	119	rspcev 3576	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 <R 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)
121	120	ex 412	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ 𝐴 → (𝑦 <R 𝐶 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
122	121	a1dd 50	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → (𝑦 <R 𝐶 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
123	118, 122	syl5 34	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → ((𝑦 ∈ R ∧ ¬ (𝐶 +R -1R) <R 𝑦) → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
124	123	expcomd 416	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (¬ (𝐶 +R -1R) <R 𝑦 → (𝑦 ∈ R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))))
125	124	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝑣 ∈ P) ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → (¬ (𝐶 +R -1R) <R 𝑦 → (𝑦 ∈ R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))))
126	116, 125	pm2.61d 179	. . . . . . . 8 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝑣 ∈ P) ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → (𝑦 ∈ R → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
127	126	ralrimiv 3127	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐴 ∧ 𝑣 ∈ P) ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → ∀𝑦 ∈ R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
128	127	ex 412	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑣 ∈ P) → (∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) → ∀𝑦 ∈ R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
129	128	adantlr 715	. . . . 5 ⊢ (((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → (∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢) → ∀𝑦 ∈ R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
130	89, 129	anim12d 609	. . . 4 ⊢ (((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → ((∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → (∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))))
131		breq1 5101	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑥 <R 𝑦 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
132	131	notbid 318	. . . . . . 7 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (¬ 𝑥 <R 𝑦 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
133	132	ralbidv 3159	. . . . . 6 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
134		breq2 5102	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑦 <R 𝑥 ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
135	134	imbi1d 341	. . . . . . 7 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧) ↔ (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
136	135	ralbidv 3159	. . . . . 6 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧) ↔ ∀𝑦 ∈ R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
137	133, 136	anbi12d 632	. . . . 5 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)) ↔ (∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))))
138	137	rspcev 3576	. . . 4 ⊢ (((𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R ∧ (∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))
139	62, 130, 138	syl6an 684	. . 3 ⊢ (((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → ((∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))))
140	139	rexlimdva 3137	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → (∃𝑣 ∈ P (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ ∀𝑤 ∈ P (𝑤<P 𝑣 → ∃𝑢 ∈ 𝐵 𝑤<P 𝑢)) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))))
141	57, 140	mpd 15	1 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  ∅c0 4285  ⟨cop 4586   class class class wbr 5098  (class class class)co 7358  [cec 8633  Pcnp 10770  1_Pc1p 10771  <_P cltp 10774   ~_R cer 10775  Rcnr 10776  0_Rc0r 10777  1_Rc1r 10778  -1_Rcm1r 10779   +_R cplr 10780   <_R cltr 10782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-oadd 8401  df-omul 8402  df-er 8635  df-ec 8637  df-qs 8641  df-ni 10783  df-pli 10784  df-mi 10785  df-lti 10786  df-plpq 10819  df-mpq 10820  df-ltpq 10821  df-enq 10822  df-nq 10823  df-erq 10824  df-plq 10825  df-mq 10826  df-1nq 10827  df-rq 10828  df-ltnq 10829  df-np 10892  df-1p 10893  df-plp 10894  df-mp 10895  df-ltp 10896  df-enr 10966  df-nr 10967  df-plr 10968  df-mr 10969  df-ltr 10970  df-0r 10971  df-1r 10972  df-m1r 10973

This theorem is referenced by:  supsr  11023