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Theorem addlocprlemlt 7904

Description: Lemma for addlocpr 7909. The 𝑄 <_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 ∈ (1^st ‘𝐴))
addlocprlem.uup	⊢ (𝜑 → 𝑈 ∈ (2^nd ‘𝐴))
addlocprlem.du	⊢ (𝜑 → 𝑈 <_Q (𝐷 +_Q 𝑃))
addlocprlem.elo	⊢ (𝜑 → 𝐸 ∈ (1^st ‘𝐵))
addlocprlem.tup	⊢ (𝜑 → 𝑇 ∈ (2^nd ‘𝐵))
addlocprlem.et	⊢ (𝜑 → 𝑇 <_Q (𝐸 +_Q 𝑃))

**Assertion**
Ref	Expression
addlocprlemlt	⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))

**Proof of Theorem addlocprlemlt**
Step	Hyp	Ref	Expression
1		addlocprlem.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ P)
2		addlocprlem.dlo	. . 3 ⊢ (𝜑 → 𝐷 ∈ (1^st ‘𝐴))
3	1, 2	jca 306	. 2 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐷 ∈ (1^st ‘𝐴)))
4		addlocprlem.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ P)
5		addlocprlem.elo	. . 3 ⊢ (𝜑 → 𝐸 ∈ (1^st ‘𝐵))
6	4, 5	jca 306	. 2 ⊢ (𝜑 → (𝐵 ∈ P ∧ 𝐸 ∈ (1^st ‘𝐵)))
7		addlocprlem.qr	. . 3 ⊢ (𝜑 → 𝑄 <_Q 𝑅)
8		ltrelnq 7738	. . . . 5 ⊢ <_Q ⊆ (Q × Q)
9	8	brel 4829	. . . 4 ⊢ (𝑄 <_Q 𝑅 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
10	9	simpld 112	. . 3 ⊢ (𝑄 <_Q 𝑅 → 𝑄 ∈ Q)
11	7, 10	syl 14	. 2 ⊢ (𝜑 → 𝑄 ∈ Q)
12		addnqprl 7902	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐷 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐸 ∈ (1^st ‘𝐵))) ∧ 𝑄 ∈ Q) → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))
13	3, 6, 11, 12	syl21anc 1277	1 ⊢ (𝜑 → (𝑄 <_Q (𝐷 +_Q 𝐸) → 𝑄 ∈ (1^st ‘(𝐴 +_P 𝐵))))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∈ wcel 2209 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 1^st c1st 6376 2^nd c2nd 6377 Qcnq 7653 +_Q cplq 7655 <_Q cltq 7658 Pcnp 7664 +_P cpp 7666

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-eprel 4436 df-id 4440 df-iord 4513 df-on 4515 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-irdg 6645 df-1o 6691 df-oadd 6695 df-omul 6696 df-er 6811 df-ec 6813 df-qs 6817 df-ni 7677 df-pli 7678 df-mi 7679 df-lti 7680 df-plpq 7717 df-mpq 7718 df-enq 7720 df-nqqs 7721 df-plqqs 7722 df-mqqs 7723 df-1nqqs 7724 df-rq 7725 df-ltnqqs 7726 df-inp 7839 df-iplp 7841

This theorem is used by: addlocprlem 7908

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