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Theorem ltexprlemrnd 7667

Description: Our constructed difference is rounded. Lemma for ltexpri 7675. (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 7663	. . . . 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 7664	. . . 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 2567	. 2 ⊢ (𝐴<_P 𝐵 → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘𝐶) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘𝐶))))
7	1	ltexprlemopu 7665	. . . . 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 7666	. . . 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 2567	. 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 1503 ∈ wcel 2164 ∀wral 2472 ∃wrex 2473 {crab 2476 ⟨cop 3622 class class class wbr 4030 ‘cfv 5255 (class class class)co 5919 1^st c1st 6193 2^nd c2nd 6194 Qcnq 7342 +_Q cplq 7344 <_Q cltq 7347 <_P cltp 7357

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-eprel 4321 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-1o 6471 df-oadd 6475 df-omul 6476 df-er 6589 df-ec 6591 df-qs 6595 df-ni 7366 df-pli 7367 df-mi 7368 df-lti 7369 df-plpq 7406 df-mpq 7407 df-enq 7409 df-nqqs 7410 df-plqqs 7411 df-mqqs 7412 df-1nqqs 7413 df-ltnqqs 7415 df-inp 7528 df-iltp 7532

This theorem is referenced by: ltexprlempr 7670

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