Theorem addlocprlemeqgt 7899

Description: Lemma for addlocpr 7903. This is a step used in both the 𝑄 = (𝐷 +_Q 𝐸) and (𝐷 +_Q 𝐸) <_Q 𝑄 cases. (Contributed by Jim Kingdon, 7-Dec-2019.)

Hypotheses

Ref	Expression
addlocprlem.a	⊢ (𝜑 → 𝐴 ∈ P)
addlocprlem.b	⊢ (𝜑 → 𝐵 ∈ P)
addlocprlem.qr	⊢ (𝜑 → 𝑄 <Q 𝑅)
addlocprlem.p	⊢ (𝜑 → 𝑃 ∈ Q)
addlocprlem.qppr	⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
addlocprlem.dlo	⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
addlocprlem.uup	⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
addlocprlem.du	⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
addlocprlem.elo	⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
addlocprlem.tup	⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))
addlocprlem.et	⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))

Assertion

Ref	Expression
addlocprlemeqgt	⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Proof of Theorem addlocprlemeqgt

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

Step	Hyp	Ref	Expression
1		addlocprlem.du	. . 3 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
2		addlocprlem.et	. . 3 ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))
3		addlocprlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ P)
4		prop 7842	. . . . . 6 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
6		addlocprlem.uup	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
7		elprnqu 7849	. . . . 5 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) → 𝑈 ∈ Q)
8	5, 6, 7	syl2anc 415	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Q)
9		addlocprlem.dlo	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
10		elprnql 7848	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
11	5, 9, 10	syl2anc 415	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Q)
13		addclnq 7742	. . . . 5 ⊢ ((𝐷 ∈ Q ∧ 𝑃 ∈ Q) → (𝐷 +Q 𝑃) ∈ Q)
14	11, 12, 13	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
16		prop 7842	. . . . . 6 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
17	15, 16	syl 14	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
18		addlocprlem.tup	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))
19		elprnqu 7849	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑇 ∈ Q)
20	17, 18, 19	syl2anc 415	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Q)
21		addlocprlem.elo	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
22		elprnql 7848	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
23	17, 21, 22	syl2anc 415	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Q)
24		addclnq 7742	. . . . 5 ⊢ ((𝐸 ∈ Q ∧ 𝑃 ∈ Q) → (𝐸 +Q 𝑃) ∈ Q)
25	23, 12, 24	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26		lt2addnq 7771	. . . 4 ⊢ (((𝑈 ∈ Q ∧ (𝐷 +Q 𝑃) ∈ Q) ∧ (𝑇 ∈ Q ∧ (𝐸 +Q 𝑃) ∈ Q)) → ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝑇 <Q (𝐸 +Q 𝑃)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃))))
27	8, 14, 20, 25, 26	syl22anc 1279	. . 3 ⊢ (𝜑 → ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝑇 <Q (𝐸 +Q 𝑃)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃))))
28	1, 2, 27	mp2and 437	. 2 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29		addcomnqg 7748	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
30	29	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
31		addassnqg 7749	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
32	31	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → ((𝑓 +Q 𝑔) +Q ℎ) = (𝑓 +Q (𝑔 +Q ℎ)))
33		addclnq 7742	. . . 4 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) ∈ Q)
34	33	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) ∈ Q)
35	11, 12, 23, 30, 32, 12, 34	caov4d 6274	. 2 ⊢ (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
36	28, 35	breqtrd 4156	1 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3712   class class class wbr 4130  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  Qcnq 7647   +_Q cplq 7649   <_Q cltq 7652  Pcnp 7658

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-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-enq 7714  df-nqqs 7715  df-plqqs 7716  df-ltnqqs 7720  df-inp 7833

This theorem is used by:  addlocprlemeq  7900  addlocprlemgt  7901