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Theorem ltexprlemrnd 7672

Description: Our constructed difference is rounded. Lemma for ltexpri 7680. (Contributed by Jim Kingdon, 17-Dec-2019.)

**Hypothesis**
Ref	Expression
ltexprlem.1	⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩

**Assertion**
Ref	Expression
ltexprlemrnd	⊢ (𝐴<_P 𝐵 → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)))))

Distinct variable groups: 𝑥,𝑦,𝑞,𝑟,𝐴 𝑥,𝐵,𝑦,𝑞,𝑟 𝑥,𝐶,𝑦,𝑞,𝑟

**Proof of Theorem ltexprlemrnd**
Step	Hyp	Ref	Expression
1		ltexprlem.1	. . . . . 6 ⊢ 𝐶 = ⟨{𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (2^nd ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (1^st ‘𝐵))}, {𝑥 ∈ Q ∣ ∃𝑦(𝑦 ∈ (1^st ‘𝐴) ∧ (𝑦 +_Q 𝑥) ∈ (2^nd ‘𝐵))}⟩
2	1	ltexprlemopl 7668	. . . . 5 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q ∧ 𝑞 ∈ (1^st ‘𝐶)) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)))
3	2	3expia 1207	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → (𝑞 ∈ (1^st ‘𝐶) → ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))))
4	1	ltexprlemlol 7669	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → (∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶)) → 𝑞 ∈ (1^st ‘𝐶)))
5	3, 4	impbid 129	. . 3 ⊢ ((𝐴<_P 𝐵 ∧ 𝑞 ∈ Q) → (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))))
6	5	ralrimiva 2570	. 2 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))))
7	1	ltexprlemopu 7670	. . . . 5 ⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q ∧ 𝑟 ∈ (2^nd ‘𝐶)) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)))
8	7	3expia 1207	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2^nd ‘𝐶) → ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶))))
9	1	ltexprlemupu 7671	. . . 4 ⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) → (∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)) → 𝑟 ∈ (2^nd ‘𝐶)))
10	8, 9	impbid 129	. . 3 ⊢ ((𝐴<_P 𝐵 ∧ 𝑟 ∈ Q) → (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶))))
11	10	ralrimiva 2570	. 2 ⊢ (𝐴<_P 𝐵 → ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶))))
12	6, 11	jca 306	1 ⊢ (𝐴<_P 𝐵 → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘𝐶) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘𝐶)))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1506 ∈ wcel 2167 ∀wral 2475 ∃wrex 2476 {crab 2479 ⟨cop 3625 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 1^st c1st 6196 2^nd c2nd 6197 Qcnq 7347 +_Q cplq 7349 <_Q cltq 7352 <_P cltp 7362

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-eprel 4324 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-1o 6474 df-oadd 6478 df-omul 6479 df-er 6592 df-ec 6594 df-qs 6598 df-ni 7371 df-pli 7372 df-mi 7373 df-lti 7374 df-plpq 7411 df-mpq 7412 df-enq 7414 df-nqqs 7415 df-plqqs 7416 df-mqqs 7417 df-1nqqs 7418 df-ltnqqs 7420 df-inp 7533 df-iltp 7537

This theorem is referenced by: ltexprlempr 7675

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