Theorem caucvgprlemcanl 7394

Description: Lemma for cauappcvgprlemladdrl 7407. 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 7369	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
2	1	adantl 273	. . 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 7297	. . . 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 7297	. . . . 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 7287	. . . 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 406	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ∈ P)
12		caucvgprlemcanl.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Q)
13		nqprlu 7297	. . . 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 7328	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
16	15	adantl 273	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
17	2, 5, 11, 14, 16	caovord2d 5892	. 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 7301	. . 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 406	. 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 7311	. . . . 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 406	. . . 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 7311	. . . . . 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 406	. . . . 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 5742	. . . 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 3906	. . 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 7125	. . . . 5 ⊢ ((𝑅 ∈ Q ∧ 𝑄 ∈ Q) → (𝑅 +Q 𝑄) ∈ Q)
27	3, 12, 26	syl2anc 406	. . . 4 ⊢ (𝜑 → (𝑅 +Q 𝑄) ∈ Q)
28		addclnq 7125	. . . . . . 7 ⊢ ((𝑆 ∈ Q ∧ 𝑄 ∈ Q) → (𝑆 +Q 𝑄) ∈ Q)
29	7, 12, 28	syl2anc 406	. . . . . 6 ⊢ (𝜑 → (𝑆 +Q 𝑄) ∈ Q)
30		nqprlu 7297	. . . . . 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 7287	. . . . 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 406	. . . 4 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
34		nqprl 7301	. . . 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 406	. . 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 7329	. . . . 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 1197	. . . 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 3905	. . 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 219	. 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 220	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 103   ↔ wb 104   ∧ w3a 943   = wceq 1312   ∈ wcel 1461  {cab 2099  ⟨cop 3494   class class class wbr 3893  ‘cfv 5079  (class class class)co 5726  1^st c1st 5988  Qcnq 7030   +_Q cplq 7032   <_Q cltq 7035  Pcnp 7041   +_P cpp 7043  <_P cltp 7045

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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216  df-iplp 7218  df-iltp 7220

This theorem is referenced by:  cauappcvgprlemladdrl  7407  caucvgprlemladdrl  7428