Theorem caucvgprlemcanl 8011

Description: Lemma for cauappcvgprlemladdrl 8024. 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 7986	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
2	1	adantl 277	. . 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 7914	. . . 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 7914	. . . . 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 7904	. . . 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 415	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ∈ P)
12		caucvgprlemcanl.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Q)
13		nqprlu 7914	. . . 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 7945	. . . 4 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
16	15	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
17	2, 5, 11, 14, 16	caovord2d 6259	. 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 7918	. . 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 415	. 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 7928	. . . . 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 415	. . . 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 7928	. . . . . 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 415	. . . . 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 6101	. . . 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 4143	. . 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 7742	. . . . 5 ⊢ ((𝑅 ∈ Q ∧ 𝑄 ∈ Q) → (𝑅 +Q 𝑄) ∈ Q)
27	3, 12, 26	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑅 +Q 𝑄) ∈ Q)
28		addclnq 7742	. . . . . . 7 ⊢ ((𝑆 ∈ Q ∧ 𝑄 ∈ Q) → (𝑆 +Q 𝑄) ∈ Q)
29	7, 12, 28	syl2anc 415	. . . . . 6 ⊢ (𝜑 → (𝑆 +Q 𝑄) ∈ Q)
30		nqprlu 7914	. . . . . 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 7904	. . . . 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 415	. . . 4 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
34		nqprl 7918	. . . 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 415	. . 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 7946	. . . . 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 1278	. . . 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 4142	. . 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 220	. 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 221	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}⟩))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  {cab 2224  ⟨cop 3712   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  Qcnq 7647   +_Q cplq 7649   <_Q cltq 7652  Pcnp 7658   +_P cpp 7660  <_P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-eprel 4434  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6807  df-ec 6809  df-qs 6813  df-ni 7671  df-pli 7672  df-mi 7673  df-lti 7674  df-plpq 7711  df-mpq 7712  df-enq 7714  df-nqqs 7715  df-plqqs 7716  df-mqqs 7717  df-1nqqs 7718  df-rq 7719  df-ltnqqs 7720  df-enq0 7791  df-nq0 7792  df-0nq0 7793  df-plq0 7794  df-mq0 7795  df-inp 7833  df-iplp 7835  df-iltp 7837

This theorem is used by:  cauappcvgprlemladdrl  8024  caucvgprlemladdrl  8045