Theorem addlocprlemeqgt 7645

Description: Lemma for addlocpr 7649. 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 7588	. . . . . 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 7595	. . . . 5 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) → 𝑈 ∈ Q)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝑈 ∈ Q)
9		addlocprlem.dlo	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
10		elprnql 7594	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) → 𝐷 ∈ Q)
11	5, 9, 10	syl2anc 411	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Q)
12		addlocprlem.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Q)
13		addclnq 7488	. . . . 5 ⊢ ((𝐷 ∈ Q ∧ 𝑃 ∈ Q) → (𝐷 +Q 𝑃) ∈ Q)
14	11, 12, 13	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐷 +Q 𝑃) ∈ Q)
15		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
16		prop 7588	. . . . . 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 7595	. . . . 5 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑇 ∈ Q)
20	17, 18, 19	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝑇 ∈ Q)
21		addlocprlem.elo	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
22		elprnql 7594	. . . . . 6 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝐸 ∈ Q)
23	17, 21, 22	syl2anc 411	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ Q)
24		addclnq 7488	. . . . 5 ⊢ ((𝐸 ∈ Q ∧ 𝑃 ∈ Q) → (𝐸 +Q 𝑃) ∈ Q)
25	23, 12, 24	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐸 +Q 𝑃) ∈ Q)
26		lt2addnq 7517	. . . 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 1251	. . 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 433	. 2 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)))
29		addcomnqg 7494	. . . 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 7495	. . . 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 7488	. . . 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 6131	. 2 ⊢ (𝜑 → ((𝐷 +Q 𝑃) +Q (𝐸 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
36	28, 35	breqtrd 4070	1 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2176  ⟨cop 3636   class class class wbr 4044  ‘cfv 5271  (class class class)co 5944  1^st c1st 6224  2^nd c2nd 6225  Qcnq 7393   +_Q cplq 7395   <_Q cltq 7398  Pcnp 7404

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-ltnqqs 7466  df-inp 7579

This theorem is referenced by:  addlocprlemeq  7646  addlocprlemgt  7647