Theorem supsrlem 10522

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 10508	. . . . . . 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 486	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 𝐶 ∈ 𝐴)
5	3, 4	eqeltrd 2890	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → (𝐶 +R 0R) ∈ 𝐴)
6		1pr 10426	. . . . . . 7 ⊢ 1P ∈ P
7	6	elexi 3460	. . . . . 6 ⊢ 1P ∈ V
8		opeq1 4763	. . . . . . . . . 10 ⊢ (𝑤 = 1P → ⟨𝑤, 1P⟩ = ⟨1P, 1P⟩)
9	8	eceq1d 8311	. . . . . . . . 9 ⊢ (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = [⟨1P, 1P⟩] ~R )
10		df-0r 10471	. . . . . . . . 9 ⊢ 0R = [⟨1P, 1P⟩] ~R
11	9, 10	eqtr4di 2851	. . . . . . . 8 ⊢ (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = 0R)
12	11	oveq2d 7151	. . . . . . 7 ⊢ (𝑤 = 1P → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R 0R))
13	12	eleq1d 2874	. . . . . 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 3618	. . . . 5 ⊢ (1P ∈ 𝐵 ↔ (𝐶 +R 0R) ∈ 𝐴)
16	5, 15	sylibr 237	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 1P ∈ 𝐵)
17	16	ne0d 4251	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 𝐵 ≠ ∅)
18		breq1 5033	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦 <R 𝑥 ↔ 𝐶 <R 𝑥))
19	18	rspccv 3568	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → 𝐶 <R 𝑥))
20		0lt1sr 10506	. . . . . . . . . . . . 13 ⊢ 0R <R 1R
21		m1r 10493	. . . . . . . . . . . . . 14 ⊢ -1R ∈ R
22		ltasr 10511	. . . . . . . . . . . . . 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 233	. . . . . . . . . . . 12 ⊢ (-1R +R 0R) <R (-1R +R 1R)
25		0idsr 10508	. . . . . . . . . . . . 13 ⊢ (-1R ∈ R → (-1R +R 0R) = -1R)
26	21, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ (-1R +R 0R) = -1R
27		m1p1sr 10503	. . . . . . . . . . . 12 ⊢ (-1R +R 1R) = 0R
28	24, 26, 27	3brtr3i 5059	. . . . . . . . . . 11 ⊢ -1R <R 0R
29		ltasr 10511	. . . . . . . . . . . 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 233	. . . . . . . . . 10 ⊢ (𝐶 +R -1R) <R (𝐶 +R 0R)
32	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐶 +R 0R) = 𝐶
33	31, 32	breqtri 5055	. . . . . . . . 9 ⊢ (𝐶 +R -1R) <R 𝐶
34		ltsosr 10505	. . . . . . . . . 10 ⊢ <R Or R
35		ltrelsr 10479	. . . . . . . . . 10 ⊢ <R ⊆ (R × R)
36	34, 35	sotri 5954	. . . . . . . . 9 ⊢ (((𝐶 +R -1R) <R 𝐶 ∧ 𝐶 <R 𝑥) → (𝐶 +R -1R) <R 𝑥)
37	33, 36	mpan 689	. . . . . . . 8 ⊢ (𝐶 <R 𝑥 → (𝐶 +R -1R) <R 𝑥)
38	1	map2psrpr 10521	. . . . . . . 8 ⊢ ((𝐶 +R -1R) <R 𝑥 ↔ ∃𝑣 ∈ P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥)
39	37, 38	sylib 221	. . . . . . 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 5034	. . . . . . . . . 10 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑦 <R 𝑥))
42	41	ralbidv 3162	. . . . . . . . 9 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥))
43	14	abeq2i 2925	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴)
44		breq1 5033	. . . . . . . . . . . . 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 3568	. . . . . . . . . . . 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 10520	. . . . . . . . . . . 12 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑤<P 𝑣)
47	45, 46	syl6ib 254	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 → 𝑤<P 𝑣))
48	43, 47	syl5bi 245	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑤 ∈ 𝐵 → 𝑤<P 𝑣))
49	48	ralrimiv 3148	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∀𝑤 ∈ 𝐵 𝑤<P 𝑣)
50	42, 49	syl6bir 257	. . . . . . . 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 3232	. . . . . 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 3241	. . . 4 ⊢ (∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣))
55	54	impcom 411	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣)
56		supexpr 10465	. . 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 587	. 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 10519	. . . . . . 7 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑣 ∈ P)
59	35	brel 5581	. . . . . . 7 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
60	58, 59	sylbir 238	. . . . . 6 ⊢ (𝑣 ∈ P → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
61	60	simprd 499	. . . . 5 ⊢ (𝑣 ∈ P → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
62	61	adantl 485	. . . 4 ⊢ (((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
63	34, 35	sotri 5954	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (𝐶 +R -1R) <R 𝑦)
65	1	map2psrpr 10521	. . . . . . . . . . . . . 14 ⊢ ((𝐶 +R -1R) <R 𝑦 ↔ ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
66	64, 65	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
67		rexex 3203	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
68		df-ral 3111	. . . . . . . . . . . . . . 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 2877	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
71	43, 70	syl5bb 286	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤 ∈ 𝐵 ↔ 𝑦 ∈ 𝐴))
72	1	ltpsrpr 10520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ↔ 𝑣<P 𝑤)
73		breq2 5034	. . . . . . . . . . . . . . . . . . . . 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 288	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑣<P 𝑤 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
75	74	notbid 321	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (¬ 𝑣<P 𝑤 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
76	71, 75	imbi12d 348	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ↔ (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
77	76	biimpac 482	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 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 417	. . . . . . . . . . . . 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 414	. . . . . . . . . . 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 455	. . . . . . . . . 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 193	. . . . . . . . 9 ⊢ ((𝑣 ∈ P ∧ (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
86	85	expr 460	. . . . . . . 8 ⊢ ((𝑣 ∈ P ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤) → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
87	86	ralrimiv 3148	. . . . . . 7 ⊢ ((𝑣 ∈ P ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤) → ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
88	87	ex 416	. . . . . 6 ⊢ (𝑣 ∈ P → (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 → ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
89	88	adantl 485	. . . . 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 3216	. . . . . . . . . . . . . 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 5033	. . . . . . . . . . . . . . . . . . 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 288	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤<P 𝑣 ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
93	92	biimprd 251	. . . . . . . . . . . . . . . . 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 4763	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑢 → ⟨𝑤, 1P⟩ = ⟨𝑢, 1P⟩)
96	95	eceq1d 8311	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑢 → [⟨𝑤, 1P⟩] ~R = [⟨𝑢, 1P⟩] ~R )
97	96	oveq2d 7151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑢 → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R [⟨𝑢, 1P⟩] ~R ))
98	97	eleq1d 2874	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑢 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴))
99	94, 98, 14	elab2 3618	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ 𝐵 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴)
100		breq2 5034	. . . . . . . . . . . . . . . . . . . . . 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 10520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ↔ 𝑤<P 𝑢)
102	100, 101	syl6bb 290	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝐶 +R [⟨𝑢, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ 𝑤<P 𝑢))
103	102	rspcev 3571	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴 ∧ 𝑤<P 𝑢) → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
104	99, 103	sylanb 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤<P 𝑢) → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
105	104	rexlimiva 3240	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑢 ∈ 𝐵 𝑤<P 𝑢 → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
106		breq1 5033	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ 𝑦 <R 𝑧))
107	106	rexbidv 3256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))
108	105, 107	syl5ib 247	. . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . . . . 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 3241	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 416	. . . . . . . . . . 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 485	. . . . . . . . . 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 5956	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ ¬ (𝐶 +R -1R) <R 𝑦 ∧ (𝐶 +R -1R) <R 𝐶) → 𝑦 <R 𝐶)
118	33, 117	mp3an3 1447	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ ¬ (𝐶 +R -1R) <R 𝑦) → 𝑦 <R 𝐶)
119		breq2 5034	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐶 → (𝑦 <R 𝑧 ↔ 𝑦 <R 𝐶))
120	119	rspcev 3571	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 <R 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)
121	120	ex 416	. . . . . . . . . . . . 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 420	. . . . . . . . . 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 725	. . . . . . . . 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 182	. . . . . . . 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 3148	. . . . . . 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 416	. . . . . 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 714	. . . . 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 611	. . . 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 5033	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑥 <R 𝑦 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
132	131	notbid 321	. . . . . . 7 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (¬ 𝑥 <R 𝑦 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
133	132	ralbidv 3162	. . . . . 6 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
134		breq2 5034	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑦 <R 𝑥 ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
135	134	imbi1d 345	. . . . . . 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 3162	. . . . . 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 633	. . . . 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 3571	. . . 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 683	. . 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 3243	. 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 209   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∅c0 4243  ⟨cop 4531   class class class wbr 5030  (class class class)co 7135  [cec 8270  Pcnp 10270  1_Pc1p 10271  <_P cltp 10274   ~_R cer 10275  Rcnr 10276  0_Rc0r 10277  1_Rc1r 10278  -1_Rcm1r 10279   +_R cplr 10280   <_R cltr 10282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-ec 8274  df-qs 8278  df-ni 10283  df-pli 10284  df-mi 10285  df-lti 10286  df-plpq 10319  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-plq 10325  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329  df-np 10392  df-1p 10393  df-plp 10394  df-mp 10395  df-ltp 10396  df-enr 10466  df-nr 10467  df-plr 10468  df-mr 10469  df-ltr 10470  df-0r 10471  df-1r 10472  df-m1r 10473

This theorem is referenced by:  supsr  10523