Theorem addnqprlemrl 6538

Description: Lemma for addnqpr 6542. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

Ref	Expression
addnqprlemrl	⊢ ((A ∈ Q ∧ B ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩))

Distinct variable groups:   A,𝑙,u   B,𝑙,u

Proof of Theorem addnqprlemrl

Dummy variables f g ℎ 𝑟 𝑠 𝑡 x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nqprlu 6530	. . . . . 6 ⊢ (A ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ ∈ P)
2		nqprlu 6530	. . . . . 6 ⊢ (B ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩ ∈ P)
3		df-iplp 6451	. . . . . . 7 ⊢ +P = (x ∈ P, y ∈ P ↦ ⟨{f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (1st ‘x) ∧ ℎ ∈ (1st ‘y) ∧ f = (g +Q ℎ))}, {f ∈ Q ∣ ∃g ∈ Q ∃ℎ ∈ Q (g ∈ (2nd ‘x) ∧ ℎ ∈ (2nd ‘y) ∧ f = (g +Q ℎ))}⟩)
4		addclnq 6359	. . . . . . 7 ⊢ ((g ∈ Q ∧ ℎ ∈ Q) → (g +Q ℎ) ∈ Q)
5	3, 4	genpelvl 6495	. . . . . 6 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩ ∈ P) → (𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)) ↔ ∃𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)𝑟 = (𝑠 +Q 𝑡)))
6	1, 2, 5	syl2an 273	. . . . 5 ⊢ ((A ∈ Q ∧ B ∈ Q) → (𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)) ↔ ∃𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)𝑟 = (𝑠 +Q 𝑡)))
7	6	biimpa 280	. . . 4 ⊢ (((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) → ∃𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)𝑟 = (𝑠 +Q 𝑡))
8		vex 2554	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
9		breq1 3758	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑠 → (𝑙 <Q A ↔ 𝑠 <Q A))
10		ltnqex 6531	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q A} ∈ V
11		gtnqex 6532	. . . . . . . . . . . . . 14 ⊢ {u ∣ A <Q u} ∈ V
12	10, 11	op1st 5715	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) = {𝑙 ∣ 𝑙 <Q A}
13	8, 9, 12	elab2 2684	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ↔ 𝑠 <Q A)
14	13	biimpi 113	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) → 𝑠 <Q A)
15	14	ad2antrl 459	. . . . . . . . . 10 ⊢ ((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) → 𝑠 <Q A)
16	15	adantr 261	. . . . . . . . 9 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑠 <Q A)
17		vex 2554	. . . . . . . . . . . . 13 ⊢ 𝑡 ∈ V
18		breq1 3758	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑡 → (𝑙 <Q B ↔ 𝑡 <Q B))
19		ltnqex 6531	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q B} ∈ V
20		gtnqex 6532	. . . . . . . . . . . . . 14 ⊢ {u ∣ B <Q u} ∈ V
21	19, 20	op1st 5715	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩) = {𝑙 ∣ 𝑙 <Q B}
22	17, 18, 21	elab2 2684	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩) ↔ 𝑡 <Q B)
23	22	biimpi 113	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩) → 𝑡 <Q B)
24	23	ad2antll 460	. . . . . . . . . 10 ⊢ ((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) → 𝑡 <Q B)
25	24	adantr 261	. . . . . . . . 9 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑡 <Q B)
26		ltrelnq 6349	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
27	26	brel 4335	. . . . . . . . . . 11 ⊢ (𝑠 <Q A → (𝑠 ∈ Q ∧ A ∈ Q))
28	16, 27	syl 14	. . . . . . . . . 10 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 ∈ Q ∧ A ∈ Q))
29	26	brel 4335	. . . . . . . . . . 11 ⊢ (𝑡 <Q B → (𝑡 ∈ Q ∧ B ∈ Q))
30	25, 29	syl 14	. . . . . . . . . 10 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑡 ∈ Q ∧ B ∈ Q))
31		lt2addnq 6388	. . . . . . . . . 10 ⊢ (((𝑠 ∈ Q ∧ A ∈ Q) ∧ (𝑡 ∈ Q ∧ B ∈ Q)) → ((𝑠 <Q A ∧ 𝑡 <Q B) → (𝑠 +Q 𝑡) <Q (A +Q B)))
32	28, 30, 31	syl2anc 391	. . . . . . . . 9 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ((𝑠 <Q A ∧ 𝑡 <Q B) → (𝑠 +Q 𝑡) <Q (A +Q B)))
33	16, 25, 32	mp2and 409	. . . . . . . 8 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑠 +Q 𝑡) <Q (A +Q B))
34		breq1 3758	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 <Q (A +Q B) ↔ (𝑠 +Q 𝑡) <Q (A +Q B)))
35	34	adantl 262	. . . . . . . 8 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝑟 <Q (A +Q B) ↔ (𝑠 +Q 𝑡) <Q (A +Q B)))
36	33, 35	mpbird 156	. . . . . . 7 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 <Q (A +Q B))
37		vex 2554	. . . . . . . 8 ⊢ 𝑟 ∈ V
38		breq1 3758	. . . . . . . 8 ⊢ (𝑙 = 𝑟 → (𝑙 <Q (A +Q B) ↔ 𝑟 <Q (A +Q B)))
39		ltnqex 6531	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q (A +Q B)} ∈ V
40		gtnqex 6532	. . . . . . . . 9 ⊢ {u ∣ (A +Q B) <Q u} ∈ V
41	39, 40	op1st 5715	. . . . . . . 8 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩) = {𝑙 ∣ 𝑙 <Q (A +Q B)}
42	37, 38, 41	elab2 2684	. . . . . . 7 ⊢ (𝑟 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩) ↔ 𝑟 <Q (A +Q B))
43	36, 42	sylibr 137	. . . . . 6 ⊢ (((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩))
44	43	ex 108	. . . . 5 ⊢ ((((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) ∧ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩) ∧ 𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩)))
45	44	rexlimdvva 2434	. . . 4 ⊢ (((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) → (∃𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩)∃𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩)))
46	7, 45	mpd 13	. . 3 ⊢ (((A ∈ Q ∧ B ∈ Q) ∧ 𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩))) → 𝑟 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩))
47	46	ex 108	. 2 ⊢ ((A ∈ Q ∧ B ∈ Q) → (𝑟 ∈ (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)) → 𝑟 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩)))
48	47	ssrdv 2945	1 ⊢ ((A ∈ Q ∧ B ∈ Q) → (1st ‘(⟨{𝑙 ∣ 𝑙 <Q A}, {u ∣ A <Q u}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q B}, {u ∣ B <Q u}⟩)) ⊆ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (A +Q B)}, {u ∣ (A +Q B) <Q u}⟩))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∃wrex 2301   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  ‘cfv 4845  (class class class)co 5455  1^st c1st 5707  Qcnq 6264   +_Q cplq 6266   <_Q cltq 6269  Pcnp 6275   +_P cpp 6277

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-iplp 6451

This theorem is referenced by:  addnqprlemfu  6541  addnqpr  6542