Theorem supsrlem 10535

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 10521	. . . . . . 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 485	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 𝐶 ∈ 𝐴)
5	3, 4	eqeltrd 2915	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → (𝐶 +R 0R) ∈ 𝐴)
6		1pr 10439	. . . . . . 7 ⊢ 1P ∈ P
7	6	elexi 3515	. . . . . 6 ⊢ 1P ∈ V
8		opeq1 4805	. . . . . . . . . 10 ⊢ (𝑤 = 1P → ⟨𝑤, 1P⟩ = ⟨1P, 1P⟩)
9	8	eceq1d 8330	. . . . . . . . 9 ⊢ (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = [⟨1P, 1P⟩] ~R )
10		df-0r 10484	. . . . . . . . 9 ⊢ 0R = [⟨1P, 1P⟩] ~R
11	9, 10	syl6eqr 2876	. . . . . . . 8 ⊢ (𝑤 = 1P → [⟨𝑤, 1P⟩] ~R = 0R)
12	11	oveq2d 7174	. . . . . . 7 ⊢ (𝑤 = 1P → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R 0R))
13	12	eleq1d 2899	. . . . . 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 3672	. . . . 5 ⊢ (1P ∈ 𝐵 ↔ (𝐶 +R 0R) ∈ 𝐴)
16	5, 15	sylibr 236	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 1P ∈ 𝐵)
17	16	ne0d 4303	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → 𝐵 ≠ ∅)
18		breq1 5071	. . . . . . . 8 ⊢ (𝑦 = 𝐶 → (𝑦 <R 𝑥 ↔ 𝐶 <R 𝑥))
19	18	rspccv 3622	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → 𝐶 <R 𝑥))
20		0lt1sr 10519	. . . . . . . . . . . . 13 ⊢ 0R <R 1R
21		m1r 10506	. . . . . . . . . . . . . 14 ⊢ -1R ∈ R
22		ltasr 10524	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (-1R +R 0R) <R (-1R +R 1R)
25		0idsr 10521	. . . . . . . . . . . . 13 ⊢ (-1R ∈ R → (-1R +R 0R) = -1R)
26	21, 25	ax-mp 5	. . . . . . . . . . . 12 ⊢ (-1R +R 0R) = -1R
27		m1p1sr 10516	. . . . . . . . . . . 12 ⊢ (-1R +R 1R) = 0R
28	24, 26, 27	3brtr3i 5097	. . . . . . . . . . 11 ⊢ -1R <R 0R
29		ltasr 10524	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝐶 +R -1R) <R (𝐶 +R 0R)
32	1, 2	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐶 +R 0R) = 𝐶
33	31, 32	breqtri 5093	. . . . . . . . 9 ⊢ (𝐶 +R -1R) <R 𝐶
34		ltsosr 10518	. . . . . . . . . 10 ⊢ <R Or R
35		ltrelsr 10492	. . . . . . . . . 10 ⊢ <R ⊆ (R × R)
36	34, 35	sotri 5989	. . . . . . . . 9 ⊢ (((𝐶 +R -1R) <R 𝐶 ∧ 𝐶 <R 𝑥) → (𝐶 +R -1R) <R 𝑥)
37	33, 36	mpan 688	. . . . . . . 8 ⊢ (𝐶 <R 𝑥 → (𝐶 +R -1R) <R 𝑥)
38	1	map2psrpr 10534	. . . . . . . 8 ⊢ ((𝐶 +R -1R) <R 𝑥 ↔ ∃𝑣 ∈ P (𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥)
39	37, 38	sylib 220	. . . . . . 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 5072	. . . . . . . . . 10 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑦 <R 𝑥))
42	41	ralbidv 3199	. . . . . . . . 9 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥))
43	14	abeq2i 2950	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝐵 ↔ (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴)
44		breq1 5071	. . . . . . . . . . . . 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 3622	. . . . . . . . . . . 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 10533	. . . . . . . . . . . 12 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑤<P 𝑣)
47	45, 46	syl6ib 253	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 → 𝑤<P 𝑣))
48	43, 47	syl5bi 244	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑤 ∈ 𝐵 → 𝑤<P 𝑣))
49	48	ralrimiv 3183	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ∀𝑤 ∈ 𝐵 𝑤<P 𝑣)
50	42, 49	syl6bir 256	. . . . . . . 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 3275	. . . . . 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 3284	. . . 4 ⊢ (∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → (𝐶 ∈ 𝐴 → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣))
55	54	impcom 410	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑣 ∈ P ∀𝑤 ∈ 𝐵 𝑤<P 𝑣)
56		supexpr 10478	. . 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 586	. 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 10532	. . . . . . 7 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ↔ 𝑣 ∈ P)
59	35	brel 5619	. . . . . . 7 ⊢ ((𝐶 +R -1R) <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
60	58, 59	sylbir 237	. . . . . 6 ⊢ (𝑣 ∈ P → ((𝐶 +R -1R) ∈ R ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R))
61	60	simprd 498	. . . . 5 ⊢ (𝑣 ∈ P → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
62	61	adantl 484	. . . 4 ⊢ (((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) ∧ 𝑣 ∈ P) → (𝐶 +R [⟨𝑣, 1P⟩] ~R ) ∈ R)
63	34, 35	sotri 5989	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → (𝐶 +R -1R) <R 𝑦)
65	1	map2psrpr 10534	. . . . . . . . . . . . . 14 ⊢ ((𝐶 +R -1R) <R 𝑦 ↔ ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
66	64, 65	sylib 220	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ P ∧ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦) → ∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
67		rexex 3242	. . . . . . . . . . . . 13 ⊢ (∃𝑤 ∈ P (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ∃𝑤(𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦)
68		df-ral 3145	. . . . . . . . . . . . . . 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 2902	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
71	43, 70	syl5bb 285	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤 ∈ 𝐵 ↔ 𝑦 ∈ 𝐴))
72	1	ltpsrpr 10533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑤, 1P⟩] ~R ) ↔ 𝑣<P 𝑤)
73		breq2 5072	. . . . . . . . . . . . . . . . . . . . 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	syl5bbr 287	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑣<P 𝑤 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
75	74	notbid 320	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (¬ 𝑣<P 𝑤 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
76	71, 75	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝑤 ∈ 𝐵 → ¬ 𝑣<P 𝑤) ↔ (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)))
77	76	biimpac 481	. . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 416	. . . . . . . . . . . . 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 413	. . . . . . . . . . 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 454	. . . . . . . . . 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 192	. . . . . . . . 9 ⊢ ((𝑣 ∈ P ∧ (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
86	85	expr 459	. . . . . . . 8 ⊢ ((𝑣 ∈ P ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤) → (𝑦 ∈ 𝐴 → ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
87	86	ralrimiv 3183	. . . . . . 7 ⊢ ((𝑣 ∈ P ∧ ∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤) → ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦)
88	87	ex 415	. . . . . 6 ⊢ (𝑣 ∈ P → (∀𝑤 ∈ 𝐵 ¬ 𝑣<P 𝑤 → ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
89	88	adantl 484	. . . . 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 3256	. . . . . . . . . . . . . 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 5071	. . . . . . . . . . . . . . . . . . 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	syl5bbr 287	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑤<P 𝑣 ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
93	92	biimprd 250	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → (𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → 𝑤<P 𝑣))
94		vex 3499	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑢 ∈ V
95		opeq1 4805	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑢 → ⟨𝑤, 1P⟩ = ⟨𝑢, 1P⟩)
96	95	eceq1d 8330	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 = 𝑢 → [⟨𝑤, 1P⟩] ~R = [⟨𝑢, 1P⟩] ~R )
97	96	oveq2d 7174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = 𝑢 → (𝐶 +R [⟨𝑤, 1P⟩] ~R ) = (𝐶 +R [⟨𝑢, 1P⟩] ~R ))
98	97	eleq1d 2899	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = 𝑢 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) ∈ 𝐴 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴))
99	94, 98, 14	elab2 3672	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 ∈ 𝐵 ↔ (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴)
100		breq2 5072	. . . . . . . . . . . . . . . . . . . . . 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 10533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R (𝐶 +R [⟨𝑢, 1P⟩] ~R ) ↔ 𝑤<P 𝑢)
102	100, 101	syl6bb 289	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝐶 +R [⟨𝑢, 1P⟩] ~R ) → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ 𝑤<P 𝑢))
103	102	rspcev 3625	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐶 +R [⟨𝑢, 1P⟩] ~R ) ∈ 𝐴 ∧ 𝑤<P 𝑢) → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
104	99, 103	sylanb 583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑢 ∈ 𝐵 ∧ 𝑤<P 𝑢) → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
105	104	rexlimiva 3283	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑢 ∈ 𝐵 𝑤<P 𝑢 → ∃𝑧 ∈ 𝐴 (𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧)
106		breq1 5071	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) = 𝑦 → ((𝐶 +R [⟨𝑤, 1P⟩] ~R ) <R 𝑧 ↔ 𝑦 <R 𝑧))
107	106	rexbidv 3299	. . . . . . . . . . . . . . . . . 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 246	. . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 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 3284	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 484	. . . . . . . . . 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 5991	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ R ∧ ¬ (𝐶 +R -1R) <R 𝑦 ∧ (𝐶 +R -1R) <R 𝐶) → 𝑦 <R 𝐶)
118	33, 117	mp3an3 1446	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ R ∧ ¬ (𝐶 +R -1R) <R 𝑦) → 𝑦 <R 𝐶)
119		breq2 5072	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝐶 → (𝑦 <R 𝑧 ↔ 𝑦 <R 𝐶))
120	119	rspcev 3625	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 <R 𝐶) → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧)
121	120	ex 415	. . . . . . . . . . . . 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 419	. . . . . . . . . 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 724	. . . . . . . . 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 181	. . . . . . . 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 3183	. . . . . . 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 415	. . . . . 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 713	. . . . 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 610	. . . 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 5071	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑥 <R 𝑦 ↔ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
132	131	notbid 320	. . . . . . 7 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (¬ 𝑥 <R 𝑦 ↔ ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
133	132	ralbidv 3199	. . . . . 6 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ (𝐶 +R [⟨𝑣, 1P⟩] ~R ) <R 𝑦))
134		breq2 5072	. . . . . . . 8 ⊢ (𝑥 = (𝐶 +R [⟨𝑣, 1P⟩] ~R ) → (𝑦 <R 𝑥 ↔ 𝑦 <R (𝐶 +R [⟨𝑣, 1P⟩] ~R )))
135	134	imbi1d 344	. . . . . . 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 3199	. . . . . 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 3625	. . . 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 682	. . 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 3286	. 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 208   ∧ wa 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2801   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  ∅c0 4293  ⟨cop 4575   class class class wbr 5068  (class class class)co 7158  [cec 8289  Pcnp 10283  1_Pc1p 10284  <_P cltp 10287   ~_R cer 10288  Rcnr 10289  0_Rc0r 10290  1_Rc1r 10291  -1_Rcm1r 10292   +_R cplr 10293   <_R cltr 10295

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-omul 8109  df-er 8291  df-ec 8293  df-qs 8297  df-ni 10296  df-pli 10297  df-mi 10298  df-lti 10299  df-plpq 10332  df-mpq 10333  df-ltpq 10334  df-enq 10335  df-nq 10336  df-erq 10337  df-plq 10338  df-mq 10339  df-1nq 10340  df-rq 10341  df-ltnq 10342  df-np 10405  df-1p 10406  df-plp 10407  df-mp 10408  df-ltp 10409  df-enr 10479  df-nr 10480  df-plr 10481  df-mr 10482  df-ltr 10483  df-0r 10484  df-1r 10485  df-m1r 10486

This theorem is referenced by:  supsr  10536