Theorem caucvgprlemcanl 7124

Description: Lemma for cauappcvgprlemladdrl 7137. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)

Hypotheses

Ref	Expression
caucvgprlemcanl.l	⊢ (𝜑 → 𝐿 ∈ P)
caucvgprlemcanl.s	⊢ (𝜑 → 𝑆 ∈ Q)
caucvgprlemcanl.r	⊢ (𝜑 → 𝑅 ∈ Q)
caucvgprlemcanl.q	⊢ (𝜑 → 𝑄 ∈ Q)

Assertion

Ref	Expression
caucvgprlemcanl	⊢ (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))

Distinct variable groups:   𝑄,𝑙,𝑢   𝑅,𝑙,𝑢   𝑆,𝑙,𝑢

Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem caucvgprlemcanl

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltaprg 7099	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
2	1	adantl 271	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
3		caucvgprlemcanl.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Q)
4		nqprlu 7027	. . . 4 ⊢ (𝑅 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ ∈ P)
5	3, 4	syl 14	. . 3 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ ∈ P)
6		caucvgprlemcanl.l	. . . 4 ⊢ (𝜑 → 𝐿 ∈ P)
7		caucvgprlemcanl.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Q)
8		nqprlu 7027	. . . . 5 ⊢ (𝑆 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ ∈ P)
9	7, 8	syl 14	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ ∈ P)
10		addclpr 7017	. . . 4 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ∈ P)
11	6, 9, 10	syl2anc 403	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ∈ P)
12		caucvgprlemcanl.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Q)
13		nqprlu 7027	. . . 4 ⊢ (𝑄 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P)
14	12, 13	syl 14	. . 3 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P)
15		addcomprg 7058	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
16	15	adantl 271	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
17	2, 5, 11, 14, 16	caovord2d 5752	. 2 ⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ↔ (⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ((𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
18		nqprl 7031	. . 3 ⊢ ((𝑅 ∈ Q ∧ (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ∈ P) → (𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))
19	3, 11, 18	syl2anc 403	. 2 ⊢ (𝜑 → (𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩)))
20		addnqpr 7041	. . . . 5 ⊢ ((𝑅 ∈ Q ∧ 𝑄 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
21	3, 12, 20	syl2anc 403	. . . 4 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
22		addnqpr 7041	. . . . . 6 ⊢ ((𝑆 ∈ Q ∧ 𝑄 ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
23	7, 12, 22	syl2anc 403	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))
24	23	oveq2d 5610	. . . 4 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) = (𝐿 +P (⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
25	21, 24	breq12d 3827	. . 3 ⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ↔ (⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P (𝐿 +P (⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))))
26		addclnq 6855	. . . . 5 ⊢ ((𝑅 ∈ Q ∧ 𝑄 ∈ Q) → (𝑅 +Q 𝑄) ∈ Q)
27	3, 12, 26	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝑅 +Q 𝑄) ∈ Q)
28		addclnq 6855	. . . . . . 7 ⊢ ((𝑆 ∈ Q ∧ 𝑄 ∈ Q) → (𝑆 +Q 𝑄) ∈ Q)
29	7, 12, 28	syl2anc 403	. . . . . 6 ⊢ (𝜑 → (𝑆 +Q 𝑄) ∈ Q)
30		nqprlu 7027	. . . . . 6 ⊢ ((𝑆 +Q 𝑄) ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ ∈ P)
31	29, 30	syl 14	. . . . 5 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ ∈ P)
32		addclpr 7017	. . . . 5 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
33	6, 31, 32	syl2anc 403	. . . 4 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
34		nqprl 7031	. . . 4 ⊢ (((𝑅 +Q 𝑄) ∈ Q ∧ (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P) → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)))
35	27, 33, 34	syl2anc 403	. . 3 ⊢ (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ ⟨{𝑙 ∣ 𝑙 <Q (𝑅 +Q 𝑄)}, {𝑢 ∣ (𝑅 +Q 𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)))
36		addassprg 7059	. . . . 5 ⊢ ((𝐿 ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩ ∈ P) → ((𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) = (𝐿 +P (⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
37	6, 9, 14, 36	syl3anc 1172	. . . 4 ⊢ (𝜑 → ((𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) = (𝐿 +P (⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
38	37	breq2d 3826	. . 3 ⊢ (𝜑 → ((⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ((𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩) ↔ (⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P (𝐿 +P (⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩))))
39	25, 35, 38	3bitr4d 218	. 2 ⊢ (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ (⟨{𝑙 ∣ 𝑙 <Q 𝑅}, {𝑢 ∣ 𝑅 <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)<P ((𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q 𝑄}, {𝑢 ∣ 𝑄 <Q 𝑢}⟩)))
40	17, 19, 39	3bitr4rd 219	1 ⊢ (𝜑 → ((𝑅 +Q 𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩)) ↔ 𝑅 ∈ (1st ‘(𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩))))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 922   = wceq 1287   ∈ wcel 1436  {cab 2071  ⟨cop 3428   class class class wbr 3814  ‘cfv 4972  (class class class)co 5594  1^st c1st 5847  Qcnq 6760   +_Q cplq 6762   <_Q cltq 6765  Pcnp 6771   +_P cpp 6773  <_P cltp 6775

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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369

This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-2o 6117  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-enq0 6904  df-nq0 6905  df-0nq0 6906  df-plq0 6907  df-mq0 6908  df-inp 6946  df-iplp 6948  df-iltp 6950

This theorem is referenced by:  cauappcvgprlemladdrl  7137  caucvgprlemladdrl  7158