Theorem recexsrlem 11128

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 11093	. . . 4 ⊢ <R ⊆ (R × R)
2	1	brel 5743	. . 3 ⊢ (0R <R 𝐴 → (0R ∈ R ∧ 𝐴 ∈ R))
3	2	simprd 494	. 2 ⊢ (0R <R 𝐴 → 𝐴 ∈ R)
4		df-nr 11081	. . 3 ⊢ R = ((P × P) / ~R )
5		breq2 5153	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 0R <R 𝐴))
6		oveq1 7426	. . . . . 6 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = (𝐴 ·R 𝑥))
7	6	eqeq1d 2727	. . . . 5 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ (𝐴 ·R 𝑥) = 1R))
8	7	rexbidv 3168	. . . 4 ⊢ ([⟨𝑦, 𝑧⟩] ~R = 𝐴 → (∃𝑥 ∈ R ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = 1R ↔ ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R))
9	5, 8	imbi12d 343	. . 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 11103	. . . . 5 ⊢ (0R <R [⟨𝑦, 𝑧⟩] ~R ↔ 𝑧<P 𝑦)
11		ltexpri 11068	. . . . 5 ⊢ (𝑧<P 𝑦 → ∃𝑤 ∈ P (𝑧 +P 𝑤) = 𝑦)
12	10, 11	sylbi 216	. . . 4 ⊢ (0R <R [⟨𝑦, 𝑧⟩] ~R → ∃𝑤 ∈ P (𝑧 +P 𝑤) = 𝑦)
13		recexpr 11076	. . . . . 6 ⊢ (𝑤 ∈ P → ∃𝑣 ∈ P (𝑤 ·P 𝑣) = 1P)
14		1pr 11040	. . . . . . . . . . . 12 ⊢ 1P ∈ P
15		addclpr 11043	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ P ∧ 1P ∈ P) → (𝑣 +P 1P) ∈ P)
16	14, 15	mpan2 689	. . . . . . . . . . 11 ⊢ (𝑣 ∈ P → (𝑣 +P 1P) ∈ P)
17		enrex 11092	. . . . . . . . . . . 12 ⊢ ~R ∈ V
18	17, 4	ecopqsi 8793	. . . . . . . . . . 11 ⊢ (((𝑣 +P 1P) ∈ P ∧ 1P ∈ P) → [⟨(𝑣 +P 1P), 1P⟩] ~R ∈ R)
19	16, 14, 18	sylancl 584	. . . . . . . . . 10 ⊢ (𝑣 ∈ P → [⟨(𝑣 +P 1P), 1P⟩] ~R ∈ R)
20	19	ad2antlr 725	. . . . . . . . 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 519	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ P → ((𝑣 +P 1P) ∈ P ∧ 1P ∈ P))
22	21	anim2i 615	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → ((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ ((𝑣 +P 1P) ∈ P ∧ 1P ∈ P)))
23	22	adantr 479	. . . . . . . . . . . 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 11101	. . . . . . . . . . . 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 7426	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 +P 𝑤) = 𝑦 → ((𝑧 +P 𝑤) ·P 𝑣) = (𝑦 ·P 𝑣))
27	26	eqcomd 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 +P 𝑤) = 𝑦 → (𝑦 ·P 𝑣) = ((𝑧 +P 𝑤) ·P 𝑣))
28		vex 3465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑧 ∈ V
29		vex 3465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑤 ∈ V
30		vex 3465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑣 ∈ V
31		mulcompr 11048	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ·P 𝑓) = (𝑓 ·P 𝑢)
32		distrpr 11053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 ·P (𝑓 +P 𝑥)) = ((𝑢 ·P 𝑓) +P (𝑢 ·P 𝑥))
33	28, 29, 30, 31, 32	caovdir 7655	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣))
34		oveq2 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ·P 𝑣) = 1P → ((𝑧 ·P 𝑣) +P (𝑤 ·P 𝑣)) = ((𝑧 ·P 𝑣) +P 1P))
35	33, 34	eqtrid 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑤 ·P 𝑣) = 1P → ((𝑧 +P 𝑤) ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
36	27, 35	sylan9eqr 2787	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ·P 𝑣) = 1P ∧ (𝑧 +P 𝑤) = 𝑦) → (𝑦 ·P 𝑣) = ((𝑧 ·P 𝑣) +P 1P))
37	36	oveq1d 7434	. . . . . . . . . . . . . . . . 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 7452	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ·P 𝑣) ∈ V
39	14	elexi 3482	. . . . . . . . . . . . . . . . . 18 ⊢ 1P ∈ V
40		ovex 7452	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ·P 1P) +P (𝑧 ·P 1P)) ∈ V
41		addcompr 11046	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 +P 𝑓) = (𝑓 +P 𝑢)
42		addasspr 11047	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 +P 𝑓) +P 𝑥) = (𝑢 +P (𝑓 +P 𝑥))
43	38, 39, 40, 41, 42	caov32 7648	. . . . . . . . . . . . . . . . 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	eqtrdi 2781	. . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . 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 11047	. . . . . . . . . . . . . . 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	eqtrdi 2781	. . . . . . . . . . . . . 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 11053	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ·P (𝑣 +P 1P)) = ((𝑦 ·P 𝑣) +P (𝑦 ·P 1P))
49	48	oveq1i 7429	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = (((𝑦 ·P 𝑣) +P (𝑦 ·P 1P)) +P (𝑧 ·P 1P))
50		addasspr 11047	. . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) = ((𝑦 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
52	51	oveq1i 7429	. . . . . . . . . . . . . 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 11053	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ·P (𝑣 +P 1P)) = ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P))
54	53	oveq2i 7430	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑦 ·P 1P) +P ((𝑧 ·P 𝑣) +P (𝑧 ·P 1P)))
55		ovex 7452	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ·P 1P) ∈ V
56		ovex 7452	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ·P 1P) ∈ V
57	55, 38, 56, 41, 42	caov12 7649	. . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) = ((𝑧 ·P 𝑣) +P ((𝑦 ·P 1P) +P (𝑧 ·P 1P)))
59	58	oveq1i 7429	. . . . . . . . . . . . . 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 2790	. . . . . . . . . . . . 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 11045	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ P ∧ (𝑣 +P 1P) ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
62	16, 61	sylan2 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ P ∧ 𝑣 ∈ P) → (𝑦 ·P (𝑣 +P 1P)) ∈ P)
63		mulclpr 11045	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ P ∧ 1P ∈ P) → (𝑧 ·P 1P) ∈ P)
64	14, 63	mpan2 689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ P → (𝑧 ·P 1P) ∈ P)
65		addclpr 11043	. . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ P ∧ 𝑣 ∈ P) ∧ 𝑧 ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
67	66	an32s 650	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → ((𝑦 ·P (𝑣 +P 1P)) +P (𝑧 ·P 1P)) ∈ P)
68		mulclpr 11045	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ P ∧ 1P ∈ P) → (𝑦 ·P 1P) ∈ P)
69	14, 68	mpan2 689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ P → (𝑦 ·P 1P) ∈ P)
70		mulclpr 11045	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ P ∧ (𝑣 +P 1P) ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
71	16, 70	sylan2 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ P ∧ 𝑣 ∈ P) → (𝑧 ·P (𝑣 +P 1P)) ∈ P)
72		addclpr 11043	. . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ P ∧ (𝑧 ∈ P ∧ 𝑣 ∈ P)) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
74	73	anassrs 466	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ P ∧ 𝑧 ∈ P) ∧ 𝑣 ∈ P) → ((𝑦 ·P 1P) +P (𝑧 ·P (𝑣 +P 1P))) ∈ P)
75	67, 74	jca 510	. . . . . . . . . . . . . 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 11043	. . . . . . . . . . . . . . . 16 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
77	14, 14, 76	mp2an 690	. . . . . . . . . . . . . . 15 ⊢ (1P +P 1P) ∈ P
78	77, 14	pm3.2i 469	. . . . . . . . . . . . . 14 ⊢ ((1P +P 1P) ∈ P ∧ 1P ∈ P)
79		enreceq 11091	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 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	imbitrrid 245	. . . . . . . . . . . 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 405	. . . . . . . . . . 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 2765	. . . . . . . . . 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 11086	. . . . . . . . . 10 ⊢ 1R = [⟨(1P +P 1P), 1P⟩] ~R
85	83, 84	eqtr4di 2783	. . . . . . . . 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 7427	. . . . . . . . . . 11 ⊢ (𝑥 = [⟨(𝑣 +P 1P), 1P⟩] ~R → ([⟨𝑦, 𝑧⟩] ~R ·R 𝑥) = ([⟨𝑦, 𝑧⟩] ~R ·R [⟨(𝑣 +P 1P), 1P⟩] ~R ))
87	86	eqeq1d 2727	. . . . . . . . . 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 3606	. . . . . . . . 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 582	. . . . . . . 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 435	. . . . . . 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 3142	. . . . . 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 3142	. . . 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 8826	. 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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059  ⟨cop 4636   class class class wbr 5149  (class class class)co 7419  [cec 8723  Pcnp 10884  1_Pc1p 10885   +_P cpp 10886   ·_P cmp 10887  <_P cltp 10888   ~_R cer 10889  Rcnr 10890  0_Rc0r 10891  1_Rc1r 10892   ·_R cmr 10895   <_R cltr 10896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-omul 8492  df-er 8725  df-ec 8727  df-qs 8731  df-ni 10897  df-pli 10898  df-mi 10899  df-lti 10900  df-plpq 10933  df-mpq 10934  df-ltpq 10935  df-enq 10936  df-nq 10937  df-erq 10938  df-plq 10939  df-mq 10940  df-1nq 10941  df-rq 10942  df-ltnq 10943  df-np 11006  df-1p 11007  df-plp 11008  df-mp 11009  df-ltp 11010  df-enr 11080  df-nr 11081  df-mr 11083  df-ltr 11084  df-0r 11085  df-1r 11086

This theorem is referenced by:  recexsr  11132