Theorem recexsrlem 10319

Description: The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
recexsrlem	⊢ (0R <R 𝐴 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R)

Distinct variable group:   𝑥,𝐴

Proof of Theorem recexsrlem

Dummy variables 𝑦 𝑧 𝑤 𝑣 𝑢 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelsr 10284	. . . 4 ⊢ <R ⊆ (R × R)
2	1	brel 5464	. . 3 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 488	. 2 ⊢ (0R <R 𝐴 → 𝐴 ∈ R)
4		df-nr 10272	. . 3 ⊢ R = ((P × P) / ~R )
5		breq2 4931	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6		oveq1 6981	. . . . . 6 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
7	6	eqeq1d 2777	. . . . 5 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
8	7	rexbidv 3239	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R))
9	5, 8	imbi12d 337	. . 3 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ((0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R) ↔ (0R <R 𝐴 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R)))
10		gt0srpr 10294	. . . . 5 ⊢ (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 𝑧<P 𝑦)
11		ltexpri 10259	. . . . 5 ⊢ (𝑧<P 𝑦 → ∃𝑤 ∈ P (𝑧 +P 𝑤) = 𝑦)
12	10, 11	sylbi 209	. . . 4 ⊢ (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤 ∈ P (𝑧 +P 𝑤) = 𝑦)
13		recexpr 10267	. . . . . 6 ⊢ (𝑤 ∈ P → ∃𝑣 ∈ P (𝑤 ·P 𝑣) = 1P)
14		1pr 10231	. . . . . . . . . . . 12 ⊢ 1P ∈ P
15		addclpr 10234	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ P ∧ 1P ∈ P) → (𝑣 +P 1P) ∈ P)
16	14, 15	mpan2 678	. . . . . . . . . . 11 ⊢ (𝑣 ∈ P → (𝑣 +P 1P) ∈ P)
17		enrex 10283	. . . . . . . . . . . 12 ⊢ ~R ∈ V
18	17, 4	ecopqsi 8150	. . . . . . . . . . 11 ⊢ (((𝑣 +P 1P) ∈ P ∧ 1P ∈ P) → [⟨(𝑣 +P 1P), 1P⟩] ~R ∈ R)
19	16, 14, 18	sylancl 577	. . . . . . . . . 10 ⊢ (𝑣 ∈ P → [⟨(𝑣 +P 1P), 1P⟩] ~R ∈ R)
20	19	ad2antlr 714	. . . . . . . . 9 ⊢ ((((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨(𝑣 +P 1P), 1P⟩] ~R ∈ R)
21	16, 14	jctir 513	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ P → ((𝑣 +P 1P) ∈ P ∧ 1P ∈ P))
22	21	anim2i 607	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → ((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1P ∈ P)))
23	22	adantr 473	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1P ∈ P)))
24		mulsrpr 10292	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R )
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R )
26		oveq1 6981	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
27	26	eqcomd 2781	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
28		vex 3415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
29		vex 3415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑤 ∈ V
30		vex 3415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑣 ∈ V
31		mulcompr 10239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ·P 𝑓) = (𝑓 ·P 𝑢)
32		distrpr 10244	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ·P (𝑓 +P 𝑥)) = ((𝑢 ·P 𝑓) +P (𝑢 ·P 𝑥))
33	28, 29, 30, 31, 32	caovdir 7196	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣))
34		oveq2 6982	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
35	33, 34	syl5eq 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
36	27, 35	sylan9eqr 2833	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
37	36	oveq1d 6989	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) = (((𝑧 ·P 𝑣) +P 1P) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))))
38		ovex 7006	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ·P 𝑣) ∈ V
39	14	elexi 3431	. . . . . . . . . . . . . . . . . 18 ⊢ 1P ∈ V
40		ovex 7006	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ V
41		addcompr 10237	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 +P 𝑓) = (𝑓 +P 𝑢)
42		addasspr 10238	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 +P 𝑓) +P 𝑥) = (𝑢 +P (𝑓 +P 𝑥))
43	38, 39, 40, 41, 42	caov32 7189	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ·P 𝑣) +P 1P) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P)
44	37, 43	syl6eq 2827	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P))
45	44	oveq1d 6989	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) = ((((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) +P 1P))
46		addasspr 10238	. . . . . . . . . . . . . . 15 ⊢ ((((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) +P 1P) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P (1P +P 1P))
47	45, 46	syl6eq 2827	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P (1P +P 1P)))
48		distrpr 10244	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P))
49	48	oveq1i 6984	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P))
50		addasspr 10238	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
51	49, 50	eqtri 2799	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
52	51	oveq1i 6984	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P 1P)
53		distrpr 10244	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))
54	53	oveq2i 6985	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
55		ovex 7006	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ·P 1P) ∈ V
56		ovex 7006	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ·P 1P) ∈ V
57	55, 38, 56, 41, 42	caov12 7190	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
58	54, 57	eqtri 2799	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
59	58	oveq1i 6984	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P)) = (((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P))) +P (1P +P 1P))
60	47, 52, 59	3eqtr4g 2836	. . . . . . . . . . . . 13 ⊢ (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P)))
61		mulclpr 10236	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
62	16, 61	sylan2 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ P ∧ 𝑣 ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
63		mulclpr 10236	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ P ∧ 1P ∈ P) → (𝑧 ·P 1P) ∈ P)
64	14, 63	mpan2 678	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ P → (𝑧 ·P 1P) ∈ P)
65		addclpr 10234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ·P (𝑣 +P 1P)) ∈ P ∧ (𝑧 ·P 1P) ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
66	62, 64, 65	syl2an 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑧 ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
67	66	an32s 639	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
68		mulclpr 10236	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
69	14, 68	mpan2 678	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
70		mulclpr 10236	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ P ∧ (𝑣 +P 1P) ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
71	16, 70	sylan2 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
72		addclpr 10234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ·P 1P) ∈ P ∧ (𝑧 ·P (𝑣 +P 1P)) ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
73	69, 71, 72	syl2an 586	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ P ∧ (𝑧 ∈ P ∧ 𝑣 ∈ P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
74	73	anassrs 460	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
75	67, 74	jca 504	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P ∧ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P))
76		addclpr 10234	. . . . . . . . . . . . . . . 16 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
77	14, 14, 76	mp2an 679	. . . . . . . . . . . . . . 15 ⊢ (1P +P 1P) ∈ P
78	77, 14	pm3.2i 463	. . . . . . . . . . . . . 14 ⊢ ((1P +P 1P) ∈ P ∧ 1P ∈ P)
79		enreceq 10282	. . . . . . . . . . . . . 14 ⊢ (((((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P ∧ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P) ∧ ((1P +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R ↔ (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P))))
80	75, 78, 79	sylancl 577	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → ([⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R ↔ (((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) +P 1P) = (((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) +P (1P +P 1P))))
81	60, 80	syl5ibr 238	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R ))
82	81	imp 398	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → [⟨((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)), ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P)))⟩] ~R = [⟨(1P +P 1P), 1P⟩] ~R )
83	25, 82	eqtrd 2811	. . . . . . . . . 10 ⊢ ((((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = [⟨(1P +P 1P), 1P⟩] ~R )
84		df-1r 10277	. . . . . . . . . 10 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
85	83, 84	syl6eqr 2829	. . . . . . . . 9 ⊢ ((((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R)
86		oveq2 6982	. . . . . . . . . . 11 ⊢ (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
87	86	eqeq1d 2777	. . . . . . . . . 10 ⊢ (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R))
88	87	rspcev 3532	. . . . . . . . 9 ⊢ (([⟨(𝑣 +P 1P), 1P⟩] ~R ∈ R ∧ ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ) = 1R) → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)
89	20, 85, 88	syl2anc 576	. . . . . . . 8 ⊢ ((((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) ∧ ((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦)) → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)
90	89	exp43 429	. . . . . . 7 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑣 ∈ P → ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))))
91	90	rexlimdv 3225	. . . . . 6 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑣 ∈ P (𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
92	13, 91	syl5 34	. . . . 5 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑤 ∈ P → ((𝑧 +P 𝑤) = 𝑦 → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R)))
93	92	rexlimdv 3225	. . . 4 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (∃𝑤 ∈ P (𝑧 +P 𝑤) = 𝑦 → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
94	12, 93	syl5 34	. . 3 ⊢ ((𝑦 ∈ P ∧ 𝑧 ∈ P) → (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R))
95	4, 9, 94	ecoptocl 8183	. 2 ⊢ (𝐴 ∈ R → (0R <R 𝐴 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R))
96	3, 95	mpcom 38	1 ⊢ (0R <R 𝐴 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2048  ∃wrex 3086  ⟨cop 4445   class class class wbr 4927  (class class class)co 6974  [cec 8083  Pcnp 10075  1_Pc1p 10076   +_P cpp 10077   ·_P cmp 10078  <_P cltp 10079   ~_R cer 10080  Rcnr 10081  0_Rc0r 10082  1_Rc1r 10083   ·_R cmr 10086   <_R cltr 10087

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277  ax-inf2 8894

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-ral 3090  df-rex 3091  df-reu 3092  df-rmo 3093  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-pss 3844  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-tp 4444  df-op 4446  df-uni 4711  df-int 4748  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-tr 5029  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7498  df-2nd 7499  df-wrecs 7747  df-recs 7809  df-rdg 7847  df-1o 7901  df-oadd 7905  df-omul 7906  df-er 8085  df-ec 8087  df-qs 8091  df-ni 10088  df-pli 10089  df-mi 10090  df-lti 10091  df-plpq 10124  df-mpq 10125  df-ltpq 10126  df-enq 10127  df-nq 10128  df-erq 10129  df-plq 10130  df-mq 10131  df-1nq 10132  df-rq 10133  df-ltnq 10134  df-np 10197  df-1p 10198  df-plp 10199  df-mp 10200  df-ltp 10201  df-enr 10271  df-nr 10272  df-mr 10274  df-ltr 10275  df-0r 10276  df-1r 10277

This theorem is referenced by:  recexsr  10323