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Theorem prarloclemcalc 7521

Description: Some calculations for prarloc 7522. (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 7473	. . . . 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 5908	. . 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 7334	. . . . . . . . . . 11 ⊢ 1_o ∈ N
9		opelxpi 4673	. . . . . . . . . . 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 7458	. . . . . . . . . . 11 ⊢ ~_Q0 ∈ V
12	11	ecelqsi 6608	. . . . . . . . . 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 7444	. . . . . . . . 9 ⊢ Q₀ = ((ω × N) / ~_Q0 )
15	13, 14	eleqtrrdi 2283	. . . . . . . 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 7460	. . . . . . . 8 ⊢ Q ⊆ Q₀
18	17, 1	sselid 3168	. . . . . . 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 7471	. . . . . . 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 7472	. . . . . 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 2266	. . . 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 7394	. . . . 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 7473	. . . 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 6541	. . . . . . . . . . . . . 14 ⊢ 2_o ∈ ω
30		2on0 6446	. . . . . . . . . . . . . 14 ⊢ 2_o ≠ ∅
31		elni 7327	. . . . . . . . . . . . . 14 ⊢ (2_o ∈ N ↔ (2_o ∈ ω ∧ 2_o ≠ ∅))
32	29, 30, 31	mpbir2an 944	. . . . . . . . . . . . 13 ⊢ 2_o ∈ N
33		nnppipi 7362	. . . . . . . . . . . . 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 4673	. . . . . . . . . . . 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 7379	. . . . . . . . . . . 12 ⊢ ~_Q ∈ V
38	37	ecelqsi 6608	. . . . . . . . . . 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 7367	. . . . . . . . . 10 ⊢ Q = ((N × N) / ~_Q )
41	39, 40	eleqtrrdi 2283	. . . . . . . . 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 7395	. . . . . . . 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 7473	. . . . . . 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 7474	. . . . . . . . 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 7457	. . . . . . . . . . 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 5907	. . . . . . . 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 2225	. . . . . . 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 5908	. . . . . 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 2226	. . . . 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 7477	. . . . . . . . . 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 1337	. . . . . . . . 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 5907	. . . . . . 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 4673	. . . . . . . . . . . 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 6608	. . . . . . . . . . 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 2265	. . . . . . . . 9 ⊢ [⟨2_o, 1_o⟩] ~_Q0 ∈ Q₀
65		distnq0r 7482	. . . . . . . . 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 1337	. . . . . . . 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 2222	. . . . . 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 5908	. . . . 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 7484	. . . . . . . . 9 ⊢ (𝑄 ∈ Q₀ → ([⟨2_o, 1_o⟩] ~_Q0 ·_Q0 𝑄) = (𝑄 +_Q0 𝑄))
71	70	oveq2d 5908	. . . . . . . 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 5908	. . . . . . 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 3168	. . . . . . 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 7470	. . . . . . . 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 7481	. . . . . . 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 1249	. . . . . 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 2225	. . . . 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 2226	. . . 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 5899	. . . . . 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 2201	. . . . 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 2233	. 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 7384	. . . . . 6 ⊢ <_Q ⊆ (Q × Q)
88	87	brel 4693	. . . . 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 7419	. . . . 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 1205	. . . 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 4040	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 1364 ∈ wcel 2160 ≠ wne 2360 ∅c0 3437 ⟨cop 3610 class class class wbr 4018 ωcom 4604 × cxp 4639 (class class class)co 5892 1_oc1o 6429 2_oc2o 6430 +_o coa 6433 [cec 6552 / cqs 6553 Ncnpi 7291 ~_Q ceq 7298 Qcnq 7299 +_Q cplq 7301 ·_Q cmq 7302 <_Q cltq 7304 ~_Q0 ceq0 7305 Q₀cnq0 7306 +_Q0 cplq0 7308 ·_Q0 cmq0 7309

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4304 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-irdg 6390 df-1o 6436 df-2o 6437 df-oadd 6440 df-omul 6441 df-er 6554 df-ec 6556 df-qs 6560 df-ni 7323 df-pli 7324 df-mi 7325 df-lti 7326 df-plpq 7363 df-mpq 7364 df-enq 7366 df-nqqs 7367 df-plqqs 7368 df-mqqs 7369 df-ltnqqs 7372 df-enq0 7443 df-nq0 7444 df-plq0 7446 df-mq0 7447

This theorem is referenced by: prarloc 7522

W3C validator