Theorem nqnq0a 6644

Description: Addition of positive fractions is equal with +_Q or +_Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)

Assertion

Ref	Expression
nqnq0a	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))

Proof of Theorem nqnq0a

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

Step	Hyp	Ref	Expression
1		nqpi 6568	. . . 4 ⊢ (𝐴 ∈ Q → ∃𝑧∃𝑤((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
2		nqpi 6568	. . . 4 ⊢ (𝐵 ∈ Q → ∃𝑣∃𝑢((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q ))
3	1, 2	anim12i 331	. . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (∃𝑧∃𝑤((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ∃𝑣∃𝑢((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
4		ee4anv 1850	. . 3 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) ↔ (∃𝑧∃𝑤((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ∃𝑣∃𝑢((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
5	3, 4	sylibr 132	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → ∃𝑧∃𝑤∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
6		oveq12 5541	. . . . . . 7 ⊢ ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q ) → (𝐴 +Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q +Q [⟨𝑣, 𝑢⟩] ~Q ))
7		mulclpi 6518	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ 𝑢 ∈ N) → (𝑧 ·N 𝑢) ∈ N)
8	7	ad2ant2rl 494	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑧 ·N 𝑢) ∈ N)
9		mulclpi 6518	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ N) → (𝑤 ·N 𝑣) ∈ N)
10	9	ad2ant2lr 493	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑤 ·N 𝑣) ∈ N)
11		addpiord 6506	. . . . . . . . . . . 12 ⊢ (((𝑧 ·N 𝑢) ∈ N ∧ (𝑤 ·N 𝑣) ∈ N) → ((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)) = ((𝑧 ·N 𝑢) +𝑜 (𝑤 ·N 𝑣)))
12	8, 10, 11	syl2anc 403	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)) = ((𝑧 ·N 𝑢) +𝑜 (𝑤 ·N 𝑣)))
13		mulpiord 6507	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ N ∧ 𝑢 ∈ N) → (𝑧 ·N 𝑢) = (𝑧 ·𝑜 𝑢))
14	13	ad2ant2rl 494	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑧 ·N 𝑢) = (𝑧 ·𝑜 𝑢))
15		mulpiord 6507	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ N ∧ 𝑣 ∈ N) → (𝑤 ·N 𝑣) = (𝑤 ·𝑜 𝑣))
16	15	ad2ant2lr 493	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑤 ·N 𝑣) = (𝑤 ·𝑜 𝑣))
17	14, 16	oveq12d 5550	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑧 ·N 𝑢) +𝑜 (𝑤 ·N 𝑣)) = ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))
18	12, 17	eqtrd 2113	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)) = ((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)))
19		mulpiord 6507	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ N ∧ 𝑢 ∈ N) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
20	19	ad2ant2l 491	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑤 ·N 𝑢) = (𝑤 ·𝑜 𝑢))
21	18, 20	opeq12d 3578	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩ = ⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩)
22	21	eceq1d 6165	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 )
23		addpipqqs 6560	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q +Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩] ~Q )
24		addclpi 6517	. . . . . . . . . . 11 ⊢ (((𝑧 ·N 𝑢) ∈ N ∧ (𝑤 ·N 𝑣) ∈ N) → ((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)) ∈ N)
25	8, 10, 24	syl2anc 403	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)) ∈ N)
26		mulclpi 6518	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ N ∧ 𝑢 ∈ N) → (𝑤 ·N 𝑢) ∈ N)
27	26	ad2ant2l 491	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝑤 ·N 𝑢) ∈ N)
28		nqnq0pi 6628	. . . . . . . . . 10 ⊢ ((((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)) ∈ N ∧ (𝑤 ·N 𝑢) ∈ N) → [⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩] ~Q )
29	25, 27, 28	syl2anc 403	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩] ~Q0 = [⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩] ~Q )
30	23, 29	eqtr4d 2116	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q +Q [⟨𝑣, 𝑢⟩] ~Q ) = [⟨((𝑧 ·N 𝑢) +N (𝑤 ·N 𝑣)), (𝑤 ·N 𝑢)⟩] ~Q0 )
31		pinn 6499	. . . . . . . . . 10 ⊢ (𝑧 ∈ N → 𝑧 ∈ ω)
32	31	anim1i 333	. . . . . . . . 9 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N) → (𝑧 ∈ ω ∧ 𝑤 ∈ N))
33		pinn 6499	. . . . . . . . . 10 ⊢ (𝑣 ∈ N → 𝑣 ∈ ω)
34	33	anim1i 333	. . . . . . . . 9 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → (𝑣 ∈ ω ∧ 𝑢 ∈ N))
35		addnnnq0 6639	. . . . . . . . 9 ⊢ (((𝑧 ∈ ω ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 )
36	32, 34, 35	syl2an 283	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ) = [⟨((𝑧 ·𝑜 𝑢) +𝑜 (𝑤 ·𝑜 𝑣)), (𝑤 ·𝑜 𝑢)⟩] ~Q0 )
37	22, 30, 36	3eqtr4d 2123	. . . . . . 7 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ([⟨𝑧, 𝑤⟩] ~Q +Q [⟨𝑣, 𝑢⟩] ~Q ) = ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
38	6, 37	sylan9eqr 2135	. . . . . 6 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 +Q 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
39		nqnq0pi 6628	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ N ∧ 𝑤 ∈ N) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
40	39	adantr 270	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨𝑧, 𝑤⟩] ~Q0 = [⟨𝑧, 𝑤⟩] ~Q )
41	40	eqeq2d 2092	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ↔ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ))
42		nqnq0pi 6628	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ N ∧ 𝑢 ∈ N) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
43	42	adantl 271	. . . . . . . . . 10 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → [⟨𝑣, 𝑢⟩] ~Q0 = [⟨𝑣, 𝑢⟩] ~Q )
44	43	eqeq2d 2092	. . . . . . . . 9 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → (𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ↔ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q ))
45	41, 44	anbi12d 456	. . . . . . . 8 ⊢ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) → ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ) ↔ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
46	45	pm5.32i 441	. . . . . . 7 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 )) ↔ (((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )))
47		oveq12 5541	. . . . . . . 8 ⊢ ((𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 ) → (𝐴 +Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
48	47	adantl 271	. . . . . . 7 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q0 ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q0 )) → (𝐴 +Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
49	46, 48	sylbir 133	. . . . . 6 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 +Q0 𝐵) = ([⟨𝑧, 𝑤⟩] ~Q0 +Q0 [⟨𝑣, 𝑢⟩] ~Q0 ))
50	38, 49	eqtr4d 2116	. . . . 5 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ (𝑣 ∈ N ∧ 𝑢 ∈ N)) ∧ (𝐴 = [⟨𝑧, 𝑤⟩] ~Q ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))
51	50	an4s 552	. . . 4 ⊢ ((((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))
52	51	exlimivv 1817	. . 3 ⊢ (∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))
53	52	exlimivv 1817	. 2 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢(((𝑧 ∈ N ∧ 𝑤 ∈ N) ∧ 𝐴 = [⟨𝑧, 𝑤⟩] ~Q ) ∧ ((𝑣 ∈ N ∧ 𝑢 ∈ N) ∧ 𝐵 = [⟨𝑣, 𝑢⟩] ~Q )) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))
54	5, 53	syl 14	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐴 +Q0 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ⟨cop 3401  ωcom 4331  (class class class)co 5532   +_𝑜 coa 6021   ·_𝑜 comu 6022  [cec 6127  Ncnpi 6462   +_N cpli 6463   ·_N cmi 6464   ~_Q ceq 6469  Qcnq 6470   +_Q cplq 6472   ~_Q0 ceq0 6476   +_Q0 cplq0 6479

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-plpq 6534  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-enq0 6614  df-nq0 6615  df-plq0 6617

This theorem is referenced by:  prarloclemlo  6684  prarloclemcalc  6692