Theorem addnqprlemru 7625

Description: Lemma for addnqpr 7628. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)

Assertion

Ref	Expression
addnqprlemru	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

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

Proof of Theorem addnqprlemru

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 7614	. . . . . 6 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
2		nqprlu 7614	. . . . . 6 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
3		df-iplp 7535	. . . . . . 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 7442	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
5	3, 4	genpelvu 7580	. . . . . 6 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P) → (𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
6	1, 2, 5	syl2an 289	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ↔ ∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡)))
7	6	biimpa 296	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → ∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡))
8		vex 2766	. . . . . . . . . . . . 13 ⊢ 𝑠 ∈ V
9		breq2 4037	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑠 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑠))
10		ltnqex 7616	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
11		gtnqex 7617	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
12	10, 11	op2nd 6205	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑢 ∣ 𝐴 <Q 𝑢}
13	8, 9, 12	elab2 2912	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑠)
14	13	biimpi 120	. . . . . . . . . . 11 ⊢ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) → 𝐴 <Q 𝑠)
15	14	ad2antrl 490	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → 𝐴 <Q 𝑠)
16	15	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝐴 <Q 𝑠)
17		vex 2766	. . . . . . . . . . . . 13 ⊢ 𝑡 ∈ V
18		breq2 4037	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑡 → (𝐵 <Q 𝑢 ↔ 𝐵 <Q 𝑡))
19		ltnqex 7616	. . . . . . . . . . . . . 14 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V
20		gtnqex 7617	. . . . . . . . . . . . . 14 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V
21	19, 20	op2nd 6205	. . . . . . . . . . . . 13 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) = {𝑢 ∣ 𝐵 <Q 𝑢}
22	17, 18, 21	elab2 2912	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ↔ 𝐵 <Q 𝑡)
23	22	biimpi 120	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) → 𝐵 <Q 𝑡)
24	23	ad2antll 491	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → 𝐵 <Q 𝑡)
25	24	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝐵 <Q 𝑡)
26		ltrelnq 7432	. . . . . . . . . . . 12 ⊢ <Q ⊆ (Q × Q)
27	26	brel 4715	. . . . . . . . . . 11 ⊢ (𝐴 <Q 𝑠 → (𝐴 ∈ Q ∧ 𝑠 ∈ Q))
28	16, 27	syl 14	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐴 ∈ Q ∧ 𝑠 ∈ Q))
29	26	brel 4715	. . . . . . . . . . 11 ⊢ (𝐵 <Q 𝑡 → (𝐵 ∈ Q ∧ 𝑡 ∈ Q))
30	25, 29	syl 14	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐵 ∈ Q ∧ 𝑡 ∈ Q))
31		lt2addnq 7471	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Q ∧ 𝑠 ∈ Q) ∧ (𝐵 ∈ Q ∧ 𝑡 ∈ Q)) → ((𝐴 <Q 𝑠 ∧ 𝐵 <Q 𝑡) → (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
32	28, 30, 31	syl2anc 411	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ((𝐴 <Q 𝑠 ∧ 𝐵 <Q 𝑡) → (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
33	16, 25, 32	mp2and 433	. . . . . . . 8 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡))
34		breq2 4037	. . . . . . . . 9 ⊢ (𝑟 = (𝑠 +Q 𝑡) → ((𝐴 +Q 𝐵) <Q 𝑟 ↔ (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
35	34	adantl 277	. . . . . . . 8 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → ((𝐴 +Q 𝐵) <Q 𝑟 ↔ (𝐴 +Q 𝐵) <Q (𝑠 +Q 𝑡)))
36	33, 35	mpbird 167	. . . . . . 7 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → (𝐴 +Q 𝐵) <Q 𝑟)
37		vex 2766	. . . . . . . 8 ⊢ 𝑟 ∈ V
38		breq2 4037	. . . . . . . 8 ⊢ (𝑢 = 𝑟 → ((𝐴 +Q 𝐵) <Q 𝑢 ↔ (𝐴 +Q 𝐵) <Q 𝑟))
39		ltnqex 7616	. . . . . . . . 9 ⊢ {𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)} ∈ V
40		gtnqex 7617	. . . . . . . . 9 ⊢ {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢} ∈ V
41	39, 40	op2nd 6205	. . . . . . . 8 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) = {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}
42	37, 38, 41	elab2 2912	. . . . . . 7 ⊢ (𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩) ↔ (𝐴 +Q 𝐵) <Q 𝑟)
43	36, 42	sylibr 134	. . . . . 6 ⊢ (((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ 𝑟 = (𝑠 +Q 𝑡)) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
44	43	ex 115	. . . . 5 ⊢ ((((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) ∧ (𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → (𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
45	44	rexlimdvva 2622	. . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → (∃𝑠 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩)∃𝑡 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)𝑟 = (𝑠 +Q 𝑡) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
46	7, 45	mpd 13	. . 3 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ 𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))
47	46	ex 115	. 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝑟 ∈ (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) → 𝑟 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩)))
48	47	ssrdv 3189	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (2nd ‘(⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ⊆ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q (𝐴 +Q 𝐵)}, {𝑢 ∣ (𝐴 +Q 𝐵) <Q 𝑢}⟩))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {cab 2182  ∃wrex 2476   ⊆ wss 3157  ⟨cop 3625   class class class wbr 4033  ‘cfv 5258  (class class class)co 5922  2^nd c2nd 6197  Qcnq 7347   +_Q cplq 7349   <_Q cltq 7352  Pcnp 7358   +_P cpp 7360

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-inp 7533  df-iplp 7535

This theorem is referenced by:  addnqprlemfl  7626  addnqpr  7628