Theorem addlocprlemeq 7535

Description: Lemma for addlocpr 7538. The 𝑄 = (𝐷 +_Q 𝐸) case. (Contributed by Jim Kingdon, 6-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
addlocprlemeq	⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Proof of Theorem addlocprlemeq

Step	Hyp	Ref	Expression
1		addlocprlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ P)
2		addlocprlem.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
3		addlocprlem.qr	. . . . . 6 ⊢ (𝜑 → 𝑄 <Q 𝑅)
4		addlocprlem.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Q)
5		addlocprlem.qppr	. . . . . 6 ⊢ (𝜑 → (𝑄 +Q (𝑃 +Q 𝑃)) = 𝑅)
6		addlocprlem.dlo	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
7		addlocprlem.uup	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
8		addlocprlem.du	. . . . . 6 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
9		addlocprlem.elo	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (1st ‘𝐵))
10		addlocprlem.tup	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ (2nd ‘𝐵))
11		addlocprlem.et	. . . . . 6 ⊢ (𝜑 → 𝑇 <Q (𝐸 +Q 𝑃))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	addlocprlemeqgt 7534	. . . . 5 ⊢ (𝜑 → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
13	12	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
14		oveq1 5885	. . . . 5 ⊢ (𝑄 = (𝐷 +Q 𝐸) → (𝑄 +Q (𝑃 +Q 𝑃)) = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
15	5, 14	sylan9req 2231	. . . 4 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 = ((𝐷 +Q 𝐸) +Q (𝑃 +Q 𝑃)))
16	13, 15	breqtrrd 4033	. . 3 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → (𝑈 +Q 𝑇) <Q 𝑅)
17	1, 7	jca 306	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)))
18	2, 10	jca 306	. . . . 5 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)))
19		ltrelnq 7367	. . . . . . . 8 ⊢ <Q ⊆ (Q × Q)
20	19	brel 4680	. . . . . . 7 ⊢ (𝑄 <Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
21	20	simprd 114	. . . . . 6 ⊢ (𝑄 <Q 𝑅 → 𝑅 ∈ Q)
22	3, 21	syl 14	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Q)
23		addnqpru 7532	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵))) ∧ 𝑅 ∈ Q) → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
24	17, 18, 22, 23	syl21anc 1237	. . . 4 ⊢ (𝜑 → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
25	24	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → ((𝑈 +Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))
26	16, 25	mpd 13	. 2 ⊢ ((𝜑 ∧ 𝑄 = (𝐷 +Q 𝐸)) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵)))
27	26	ex 115	1 ⊢ (𝜑 → (𝑄 = (𝐷 +Q 𝐸) → 𝑅 ∈ (2nd ‘(𝐴 +P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   class class class wbr 4005  ‘cfv 5218  (class class class)co 5878  1^st c1st 6142  2^nd c2nd 6143  Qcnq 7282   +_Q cplq 7284   <_Q cltq 7287  Pcnp 7293   +_P cpp 7295

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-1o 6420  df-oadd 6424  df-omul 6425  df-er 6538  df-ec 6540  df-qs 6544  df-ni 7306  df-pli 7307  df-mi 7308  df-lti 7309  df-plpq 7346  df-mpq 7347  df-enq 7349  df-nqqs 7350  df-plqqs 7351  df-mqqs 7352  df-1nqqs 7353  df-rq 7354  df-ltnqqs 7355  df-inp 7468  df-iplp 7470

This theorem is referenced by:  addlocprlem  7537