Theorem caucvgprlemcanl 7764

Description: Lemma for cauappcvgprlemladdrl 7777. 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 7739	. . . 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 7667	. . . 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 7667	. . . . 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 7657	. . . 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 411	. . 3 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q 𝑆}, {𝑢 ∣ 𝑆 <Q 𝑢}⟩) ∈ P)
12		caucvgprlemcanl.q	. . . 4 ⊢ (𝜑 → 𝑄 ∈ Q)
13		nqprlu 7667	. . . 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 7698	. . . 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 6123	. 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 7671	. . 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 411	. 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 7681	. . . . 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 411	. . . 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 7681	. . . . . 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 411	. . . . 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 5967	. . . 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 4060	. . 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 7495	. . . . 5 ⊢ ((𝑅 ∈ Q ∧ 𝑄 ∈ Q) → (𝑅 +Q 𝑄) ∈ Q)
27	3, 12, 26	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑅 +Q 𝑄) ∈ Q)
28		addclnq 7495	. . . . . . 7 ⊢ ((𝑆 ∈ Q ∧ 𝑄 ∈ Q) → (𝑆 +Q 𝑄) ∈ Q)
29	7, 12, 28	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝑆 +Q 𝑄) ∈ Q)
30		nqprlu 7667	. . . . . 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 7657	. . . . 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 411	. . . 4 ⊢ (𝜑 → (𝐿 +P ⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q 𝑄)}, {𝑢 ∣ (𝑆 +Q 𝑄) <Q 𝑢}⟩) ∈ P)
34		nqprl 7671	. . . 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 411	. . 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 7699	. . . . 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 1250	. . . 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 4059	. . 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}⟩))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  {cab 2192  ⟨cop 3637   class class class wbr 4047  ‘cfv 5276  (class class class)co 5951  1^st c1st 6231  Qcnq 7400   +_Q cplq 7402   <_Q cltq 7405  Pcnp 7411   +_P cpp 7413  <_P cltp 7415

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-eprel 4340  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-1o 6509  df-2o 6510  df-oadd 6513  df-omul 6514  df-er 6627  df-ec 6629  df-qs 6633  df-ni 7424  df-pli 7425  df-mi 7426  df-lti 7427  df-plpq 7464  df-mpq 7465  df-enq 7467  df-nqqs 7468  df-plqqs 7469  df-mqqs 7470  df-1nqqs 7471  df-rq 7472  df-ltnqqs 7473  df-enq0 7544  df-nq0 7545  df-0nq0 7546  df-plq0 7547  df-mq0 7548  df-inp 7586  df-iplp 7588  df-iltp 7590

This theorem is referenced by:  cauappcvgprlemladdrl  7777  caucvgprlemladdrl  7798