Theorem addnqprlemrl 7213

Description: Lemma for addnqpr 7217. 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 7203	. . . . . 6 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
2		nqprlu 7203	. . . . . 6 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
3		df-iplp 7124	. . . . . . 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 7031	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
5	3, 4	genpelvl 7168	. . . . . 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 284	. . . . 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 291	. . . 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 2636	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
9		breq1 3870	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑠 → (𝑙 <Q 𝐴 ↔ 𝑠 <Q 𝐴))
10		ltnqex 7205	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
11		gtnqex 7206	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
12	10, 11	op1st 5955	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝐴}
13	8, 9, 12	elab2 2777	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝑠 <Q 𝐴)
14	13	biimpi 119	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝑠 <Q 𝐴)
15	14	ad2antrl 475	. . . . . . . . . 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 271	. . . . . . . . 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 2636	. . . . . . . . . . . . 13 ⊢ 𝑡 ∈ V
18		breq1 3870	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑡 → (𝑙 <Q 𝐵 ↔ 𝑡 <Q 𝐵))
19		ltnqex 7205	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V
20		gtnqex 7206	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V
21	19, 20	op1st 5955	. . . . . . . . . . . . 13 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝐵}
22	17, 18, 21	elab2 2777	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ↔ 𝑡 <Q 𝐵)
23	22	biimpi 119	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) → 𝑡 <Q 𝐵)
24	23	ad2antll 476	. . . . . . . . . 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 271	. . . . . . . . 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 7021	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
27	26	brel 4519	. . . . . . . . . . 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 4519	. . . . . . . . . . 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 7060	. . . . . . . . . 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 404	. . . . . . . . 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 425	. . . . . . . 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 3870	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 +Q 𝑡) → (𝑟 <Q (𝐴 +Q 𝐵) ↔ (𝑠 +Q 𝑡) <Q (𝐴 +Q 𝐵)))
35	34	adantl 272	. . . . . . . 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 166	. . . . . . 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 2636	. . . . . . . 8 ⊢ 𝑟 ∈ V
38		breq1 3870	. . . . . . . 8 ⊢ (𝑙 = 𝑟 → (𝑙 <Q (𝐴 +Q 𝐵) ↔ 𝑟 <Q (𝐴 +Q 𝐵)))
39		ltnqex 7205	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
40		gtnqex 7206	. . . . . . . . 9 ⊢ {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
41	39, 40	op1st 5955	. . . . . . . 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 2777	. . . . . . 7 ⊢ (𝑟 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ 𝑟 <Q (𝐴 +Q 𝐵))
43	36, 42	sylibr 133	. . . . . 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 114	. . . . 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 2510	. . . 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 114	. 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 3045	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 103   ↔ wb 104   = wceq 1296   ∈ wcel 1445  {cab 2081  ∃wrex 2371   ⊆ wss 3013  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1^st c1st 5947  Qcnq 6936   +_Q cplq 6938   <_Q cltq 6941  Pcnp 6947   +_P cpp 6949

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431

This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-eprel 4140  df-id 4144  df-po 4147  df-iso 4148  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-recs 6108  df-irdg 6173  df-1o 6219  df-oadd 6223  df-omul 6224  df-er 6332  df-ec 6334  df-qs 6338  df-ni 6960  df-pli 6961  df-mi 6962  df-lti 6963  df-plpq 7000  df-mpq 7001  df-enq 7003  df-nqqs 7004  df-plqqs 7005  df-mqqs 7006  df-1nqqs 7007  df-rq 7008  df-ltnqqs 7009  df-inp 7122  df-iplp 7124

This theorem is referenced by:  addnqprlemfu  7216  addnqpr  7217