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Theorem prarloclemcalc 7514

Description: Some calculations for prarloc 7515. (Contributed by Jim Kingdon, 26-Oct-2019.)

**Assertion**
Ref	Expression
prarloclemcalc	⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 <_Q (𝐴 +_Q 𝑃))

**Proof of Theorem prarloclemcalc**
Step	Hyp	Ref	Expression
1		simprll 537	. . . . 5 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝑄 ∈ Q)
2		nqnq0a 7466	. . . . 5 ⊢ ((𝑄 ∈ Q ∧ 𝑄 ∈ Q) → (𝑄 +_Q 𝑄) = (𝑄 +_Q0 𝑄))
3	1, 1, 2	syl2anc 411	. . . 4 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑄 +_Q 𝑄) = (𝑄 +_Q0 𝑄))
4	3	oveq2d 5904	. . 3 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝐴 +_Q0 (𝑄 +_Q 𝑄)) = (𝐴 +_Q0 (𝑄 +_Q0 𝑄)))
5		simpll 527	. . . . 5 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)))
6		simprrl 539	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝑋 ∈ Q)
7		simprrr 540	. . . . . . . 8 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝑀 ∈ ω)
8		1pi 7327	. . . . . . . . . . 11 ⊢ 1_o ∈ N
9		opelxpi 4670	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ω ∧ 1_o ∈ N) → ⟨𝑀, 1_o⟩ ∈ (ω × N))
10	8, 9	mpan2 425	. . . . . . . . . 10 ⊢ (𝑀 ∈ ω → ⟨𝑀, 1_o⟩ ∈ (ω × N))
11		enq0ex 7451	. . . . . . . . . . 11 ⊢ ~_Q0 ∈ V
12	11	ecelqsi 6602	. . . . . . . . . 10 ⊢ (⟨𝑀, 1_o⟩ ∈ (ω × N) → [⟨𝑀, 1_o⟩] ~_Q0 ∈ ((ω × N) / ~_Q0 ))
13	10, 12	syl 14	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → [⟨𝑀, 1_o⟩] ~_Q0 ∈ ((ω × N) / ~_Q0 ))
14		df-nq0 7437	. . . . . . . . 9 ⊢ Q₀ = ((ω × N) / ~_Q0 )
15	13, 14	eleqtrrdi 2281	. . . . . . . 8 ⊢ (𝑀 ∈ ω → [⟨𝑀, 1_o⟩] ~_Q0 ∈ Q₀)
16	7, 15	syl 14	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → [⟨𝑀, 1_o⟩] ~_Q0 ∈ Q₀)
17		nqnq0 7453	. . . . . . . 8 ⊢ Q ⊆ Q₀
18	17, 1	sselid 3165	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝑄 ∈ Q₀)
19		mulclnq0 7464	. . . . . . 7 ⊢ (([⟨𝑀, 1_o⟩] ~_Q0 ∈ Q₀ ∧ 𝑄 ∈ Q₀) → ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) ∈ Q₀)
20	16, 18, 19	syl2anc 411	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) ∈ Q₀)
21		nqpnq0nq 7465	. . . . . 6 ⊢ ((𝑋 ∈ Q ∧ ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) ∈ Q₀) → (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∈ Q)
22	6, 20, 21	syl2anc 411	. . . . 5 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∈ Q)
23	5, 22	eqeltrd 2264	. . . 4 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐴 ∈ Q)
24		addclnq 7387	. . . . 5 ⊢ ((𝑄 ∈ Q ∧ 𝑄 ∈ Q) → (𝑄 +_Q 𝑄) ∈ Q)
25	1, 1, 24	syl2anc 411	. . . 4 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑄 +_Q 𝑄) ∈ Q)
26		nqnq0a 7466	. . . 4 ⊢ ((𝐴 ∈ Q ∧ (𝑄 +_Q 𝑄) ∈ Q) → (𝐴 +_Q (𝑄 +_Q 𝑄)) = (𝐴 +_Q0 (𝑄 +_Q 𝑄)))
27	23, 25, 26	syl2anc 411	. . 3 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝐴 +_Q (𝑄 +_Q 𝑄)) = (𝐴 +_Q0 (𝑄 +_Q 𝑄)))
28		simplr 528	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄)))
29		2onn 6535	. . . . . . . . . . . . . 14 ⊢ 2_o ∈ ω
30		2on0 6440	. . . . . . . . . . . . . 14 ⊢ 2_o ≠ ∅
31		elni 7320	. . . . . . . . . . . . . 14 ⊢ (2_o ∈ N ↔ (2_o ∈ ω ∧ 2_o ≠ ∅))
32	29, 30, 31	mpbir2an 943	. . . . . . . . . . . . 13 ⊢ 2_o ∈ N
33		nnppipi 7355	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ ω ∧ 2_o ∈ N) → (𝑀 +_o 2_o) ∈ N)
34	32, 33	mpan2 425	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ω → (𝑀 +_o 2_o) ∈ N)
35		opelxpi 4670	. . . . . . . . . . . 12 ⊢ (((𝑀 +_o 2_o) ∈ N ∧ 1_o ∈ N) → ⟨(𝑀 +_o 2_o), 1_o⟩ ∈ (N × N))
36	34, 8, 35	sylancl 413	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ω → ⟨(𝑀 +_o 2_o), 1_o⟩ ∈ (N × N))
37		enqex 7372	. . . . . . . . . . . 12 ⊢ ~_Q ∈ V
38	37	ecelqsi 6602	. . . . . . . . . . 11 ⊢ (⟨(𝑀 +_o 2_o), 1_o⟩ ∈ (N × N) → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ∈ ((N × N) / ~_Q ))
39	36, 38	syl 14	. . . . . . . . . 10 ⊢ (𝑀 ∈ ω → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ∈ ((N × N) / ~_Q ))
40		df-nqqs 7360	. . . . . . . . . 10 ⊢ Q = ((N × N) / ~_Q )
41	39, 40	eleqtrrdi 2281	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ∈ Q)
42	7, 41	syl 14	. . . . . . . 8 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ∈ Q)
43		mulclnq 7388	. . . . . . . 8 ⊢ (([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ∈ Q ∧ 𝑄 ∈ Q) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄) ∈ Q)
44	42, 1, 43	syl2anc 411	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄) ∈ Q)
45		nqnq0a 7466	. . . . . . 7 ⊢ ((𝑋 ∈ Q ∧ ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄) ∈ Q) → (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄)) = (𝑋 +_Q0 ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄)))
46	6, 44, 45	syl2anc 411	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄)) = (𝑋 +_Q0 ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄)))
47		nqnq0m 7467	. . . . . . . . 9 ⊢ (([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ∈ Q ∧ 𝑄 ∈ Q) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄) = ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q0 𝑄))
48	42, 1, 47	syl2anc 411	. . . . . . . 8 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄) = ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q0 𝑄))
49		nqnq0pi 7450	. . . . . . . . . . 11 ⊢ (((𝑀 +_o 2_o) ∈ N ∧ 1_o ∈ N) → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 = [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q )
50	34, 8, 49	sylancl 413	. . . . . . . . . 10 ⊢ (𝑀 ∈ ω → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 = [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q )
51	7, 50	syl 14	. . . . . . . . 9 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 = [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q )
52	51	oveq1d 5903	. . . . . . . 8 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 ·_Q0 𝑄) = ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q0 𝑄))
53	48, 52	eqtr4d 2223	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄) = ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 ·_Q0 𝑄))
54	53	oveq2d 5904	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑋 +_Q0 ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄)) = (𝑋 +_Q0 ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 ·_Q0 𝑄)))
55	28, 46, 54	3eqtrd 2224	. . . . 5 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 = (𝑋 +_Q0 ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 ·_Q0 𝑄)))
56		nnanq0 7470	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ω ∧ 2_o ∈ ω ∧ 1_o ∈ N) → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 = ([⟨𝑀, 1_o⟩] ~_Q0 +_Q0 [⟨2_o, 1_o⟩] ~_Q0 ))
57	8, 56	mp3an3 1336	. . . . . . . . 9 ⊢ ((𝑀 ∈ ω ∧ 2_o ∈ ω) → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 = ([⟨𝑀, 1_o⟩] ~_Q0 +_Q0 [⟨2_o, 1_o⟩] ~_Q0 ))
58	7, 29, 57	sylancl 413	. . . . . . . 8 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → [⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 = ([⟨𝑀, 1_o⟩] ~_Q0 +_Q0 [⟨2_o, 1_o⟩] ~_Q0 ))
59	58	oveq1d 5903	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 ·_Q0 𝑄) = (([⟨𝑀, 1_o⟩] ~_Q0 +_Q0 [⟨2_o, 1_o⟩] ~_Q0 ) ·_Q0 𝑄))
60		opelxpi 4670	. . . . . . . . . . . 12 ⊢ ((2_o ∈ ω ∧ 1_o ∈ N) → ⟨2_o, 1_o⟩ ∈ (ω × N))
61	29, 8, 60	mp2an 426	. . . . . . . . . . 11 ⊢ ⟨2_o, 1_o⟩ ∈ (ω × N)
62	11	ecelqsi 6602	. . . . . . . . . . 11 ⊢ (⟨2_o, 1_o⟩ ∈ (ω × N) → [⟨2_o, 1_o⟩] ~_Q0 ∈ ((ω × N) / ~_Q0 ))
63	61, 62	ax-mp 5	. . . . . . . . . 10 ⊢ [⟨2_o, 1_o⟩] ~_Q0 ∈ ((ω × N) / ~_Q0 )
64	63, 14	eleqtrri 2263	. . . . . . . . 9 ⊢ [⟨2_o, 1_o⟩] ~_Q0 ∈ Q₀
65		distnq0r 7475	. . . . . . . . 9 ⊢ ((𝑄 ∈ Q₀ ∧ [⟨𝑀, 1_o⟩] ~_Q0 ∈ Q₀ ∧ [⟨2_o, 1_o⟩] ~_Q0 ∈ Q₀) → (([⟨𝑀, 1_o⟩] ~_Q0 +_Q0 [⟨2_o, 1_o⟩] ~_Q0 ) ·_Q0 𝑄) = (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄)))
66	64, 65	mp3an3 1336	. . . . . . . 8 ⊢ ((𝑄 ∈ Q₀ ∧ [⟨𝑀, 1_o⟩] ~_Q0 ∈ Q₀) → (([⟨𝑀, 1_o⟩] ~_Q0 +_Q0 [⟨2_o, 1_o⟩] ~_Q0 ) ·_Q0 𝑄) = (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄)))
67	18, 16, 66	syl2anc 411	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (([⟨𝑀, 1_o⟩] ~_Q0 +_Q0 [⟨2_o, 1_o⟩] ~_Q0 ) ·_Q0 𝑄) = (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄)))
68	59, 67	eqtrd 2220	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 ·_Q0 𝑄) = (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄)))
69	68	oveq2d 5904	. . . . 5 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑋 +_Q0 ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q0 ·_Q0 𝑄)) = (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄))))
70		nq02m 7477	. . . . . . . . 9 ⊢ (𝑄 ∈ Q₀ → ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄) = (𝑄 +_Q0 𝑄))
71	70	oveq2d 5904	. . . . . . . 8 ⊢ (𝑄 ∈ Q₀ → (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄)) = (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 (𝑄 +_Q0 𝑄)))
72	71	oveq2d 5904	. . . . . . 7 ⊢ (𝑄 ∈ Q₀ → (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄))) = (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 (𝑄 +_Q0 𝑄))))
73	18, 72	syl 14	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄))) = (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 (𝑄 +_Q0 𝑄))))
74	17, 6	sselid 3165	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝑋 ∈ Q₀)
75		addclnq0 7463	. . . . . . . 8 ⊢ ((𝑄 ∈ Q₀ ∧ 𝑄 ∈ Q₀) → (𝑄 +_Q0 𝑄) ∈ Q₀)
76	18, 18, 75	syl2anc 411	. . . . . . 7 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑄 +_Q0 𝑄) ∈ Q₀)
77		addassnq0 7474	. . . . . . 7 ⊢ ((𝑋 ∈ Q₀ ∧ ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) ∈ Q₀ ∧ (𝑄 +_Q0 𝑄) ∈ Q₀) → ((𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) +_Q0 (𝑄 +_Q0 𝑄)) = (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 (𝑄 +_Q0 𝑄))))
78	74, 20, 76, 77	syl3anc 1248	. . . . . 6 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ((𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) +_Q0 (𝑄 +_Q0 𝑄)) = (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 (𝑄 +_Q0 𝑄))))
79	73, 78	eqtr4d 2223	. . . . 5 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑋 +_Q0 (([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄) +_Q0 ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄))) = ((𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) +_Q0 (𝑄 +_Q0 𝑄)))
80	55, 69, 79	3eqtrd 2224	. . . 4 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 = ((𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) +_Q0 (𝑄 +_Q0 𝑄)))
81		oveq1 5895	. . . . . 6 ⊢ (𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) → (𝐴 +_Q0 (𝑄 +_Q0 𝑄)) = ((𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) +_Q0 (𝑄 +_Q0 𝑄)))
82	81	eqeq2d 2199	. . . . 5 ⊢ (𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) → (𝐵 = (𝐴 +_Q0 (𝑄 +_Q0 𝑄)) ↔ 𝐵 = ((𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) +_Q0 (𝑄 +_Q0 𝑄))))
83	5, 82	syl 14	. . . 4 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝐵 = (𝐴 +_Q0 (𝑄 +_Q0 𝑄)) ↔ 𝐵 = ((𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) +_Q0 (𝑄 +_Q0 𝑄))))
84	80, 83	mpbird 167	. . 3 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 = (𝐴 +_Q0 (𝑄 +_Q0 𝑄)))
85	4, 27, 84	3eqtr4rd 2231	. 2 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 = (𝐴 +_Q (𝑄 +_Q 𝑄)))
86		simprlr 538	. . 3 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝑄 +_Q 𝑄) <_Q 𝑃)
87		ltrelnq 7377	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
88	87	brel 4690	. . . . 5 ⊢ ((𝑄 +_Q 𝑄) <_Q 𝑃 → ((𝑄 +_Q 𝑄) ∈ Q ∧ 𝑃 ∈ Q))
89	86, 88	syl 14	. . . 4 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ((𝑄 +_Q 𝑄) ∈ Q ∧ 𝑃 ∈ Q))
90		ltanqg 7412	. . . . 5 ⊢ (((𝑄 +_Q 𝑄) ∈ Q ∧ 𝑃 ∈ Q ∧ 𝐴 ∈ Q) → ((𝑄 +_Q 𝑄) <_Q 𝑃 ↔ (𝐴 +_Q (𝑄 +_Q 𝑄)) <_Q (𝐴 +_Q 𝑃)))
91	90	3expa 1204	. . . 4 ⊢ ((((𝑄 +_Q 𝑄) ∈ Q ∧ 𝑃 ∈ Q) ∧ 𝐴 ∈ Q) → ((𝑄 +_Q 𝑄) <_Q 𝑃 ↔ (𝐴 +_Q (𝑄 +_Q 𝑄)) <_Q (𝐴 +_Q 𝑃)))
92	89, 23, 91	syl2anc 411	. . 3 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → ((𝑄 +_Q 𝑄) <_Q 𝑃 ↔ (𝐴 +_Q (𝑄 +_Q 𝑄)) <_Q (𝐴 +_Q 𝑃)))
93	86, 92	mpbid 147	. 2 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → (𝐴 +_Q (𝑄 +_Q 𝑄)) <_Q (𝐴 +_Q 𝑃))
94	85, 93	eqbrtrd 4037	1 ⊢ (((𝐴 = (𝑋 +_Q0 ([⟨𝑀, 1_o⟩] ~_Q0 ·_Q0 𝑄)) ∧ 𝐵 = (𝑋 +_Q ([⟨(𝑀 +_o 2_o), 1_o⟩] ~_Q ·_Q 𝑄))) ∧ ((𝑄 ∈ Q ∧ (𝑄 +_Q 𝑄) <_Q 𝑃) ∧ (𝑋 ∈ Q ∧ 𝑀 ∈ ω))) → 𝐵 <_Q (𝐴 +_Q 𝑃))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ≠ wne 2357 ∅c0 3434 ⟨cop 3607 class class class wbr 4015 ωcom 4601 × cxp 4636 (class class class)co 5888 1_oc1o 6423 2_oc2o 6424 +_o coa 6427 [cec 6546 / cqs 6547 Ncnpi 7284 ~_Q ceq 7291 Qcnq 7292 +_Q cplq 7294 ·_Q cmq 7295 <_Q cltq 7297 ~_Q0 ceq0 7298 Q₀cnq0 7299 +_Q0 cplq0 7301 ·_Q0 cmq0 7302

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-eprel 4301 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-1o 6430 df-2o 6431 df-oadd 6434 df-omul 6435 df-er 6548 df-ec 6550 df-qs 6554 df-ni 7316 df-pli 7317 df-mi 7318 df-lti 7319 df-plpq 7356 df-mpq 7357 df-enq 7359 df-nqqs 7360 df-plqqs 7361 df-mqqs 7362 df-ltnqqs 7365 df-enq0 7436 df-nq0 7437 df-plq0 7439 df-mq0 7440

This theorem is referenced by: prarloc 7515

W3C validator