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Theorem ltexpri 6803

Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)

**Assertion**
Ref	Expression
ltexpri	⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

Proof of Theorem ltexpri Dummy variables 𝑦 𝑧 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 108	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑧 = 𝑣)
2	1	eleq1d 2147	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (2^nd ‘𝐴) ↔ 𝑣 ∈ (2^nd ‘𝐴)))
3		simpl 107	. . . . . . . . 9 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → 𝑦 = 𝑢)
4	1, 3	oveq12d 5550	. . . . . . . 8 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 +_Q 𝑦) = (𝑣 +_Q 𝑢))
5	4	eleq1d 2147	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵)))
6	2, 5	anbi12d 456	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ (𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
7	6	cbvexdva 1845	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))))
8	7	cbvrabv 2600	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}
9	1	eleq1d 2147	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → (𝑧 ∈ (1^st ‘𝐴) ↔ 𝑣 ∈ (1^st ‘𝐴)))
10	4	eleq1d 2147	. . . . . . 7 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵) ↔ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵)))
11	9, 10	anbi12d 456	. . . . . 6 ⊢ ((𝑦 = 𝑢 ∧ 𝑧 = 𝑣) → ((𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ (𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
12	11	cbvexdva 1845	. . . . 5 ⊢ (𝑦 = 𝑢 → (∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵)) ↔ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))))
13	12	cbvrabv 2600	. . . 4 ⊢ {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))} = {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}
14	8, 13	opeq12i 3575	. . 3 ⊢ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ = ⟨{𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (2^nd ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (1^st ‘𝐵))}, {𝑢 ∈ Q ∣ ∃𝑣(𝑣 ∈ (1^st ‘𝐴) ∧ (𝑣 +_Q 𝑢) ∈ (2^nd ‘𝐵))}⟩
15	14	ltexprlempr 6798	. 2 ⊢ (𝐴<_P 𝐵 → ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P)
16	14	ltexprlemfl 6799	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (1^st ‘𝐵))
17	14	ltexprlemrl 6800	. . . 4 ⊢ (𝐴<_P 𝐵 → (1^st ‘𝐵) ⊆ (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
18	16, 17	eqssd 3016	. . 3 ⊢ (𝐴<_P 𝐵 → (1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵))
19	14	ltexprlemfu 6801	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) ⊆ (2^nd ‘𝐵))
20	14	ltexprlemru 6802	. . . 4 ⊢ (𝐴<_P 𝐵 → (2^nd ‘𝐵) ⊆ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)))
21	19, 20	eqssd 3016	. . 3 ⊢ (𝐴<_P 𝐵 → (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))
22		ltrelpr 6695	. . . . . . 7 ⊢ <_P ⊆ (P × P)
23	22	brel 4410	. . . . . 6 ⊢ (𝐴<_P 𝐵 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
24	23	simpld 110	. . . . 5 ⊢ (𝐴<_P 𝐵 → 𝐴 ∈ P)
25		addclpr 6727	. . . . 5 ⊢ ((𝐴 ∈ P ∧ ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P) → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
26	24, 15, 25	syl2anc 403	. . . 4 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P)
27	23	simprd 112	. . . 4 ⊢ (𝐴<_P 𝐵 → 𝐵 ∈ P)
28		preqlu 6662	. . . 4 ⊢ (((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
29	26, 27, 28	syl2anc 403	. . 3 ⊢ (𝐴<_P 𝐵 → ((𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵 ↔ ((1^st ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (1^st ‘𝐵) ∧ (2^nd ‘(𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩)) = (2^nd ‘𝐵))))
30	18, 21, 29	mpbir2and 885	. 2 ⊢ (𝐴<_P 𝐵 → (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵)
31		oveq2 5540	. . . 4 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → (𝐴 +_P 𝑥) = (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩))
32	31	eqeq1d 2089	. . 3 ⊢ (𝑥 = ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ → ((𝐴 +_P 𝑥) = 𝐵 ↔ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵))
33	32	rspcev 2701	. 2 ⊢ ((⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩ ∈ P ∧ (𝐴 +_P ⟨{𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (2^nd ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (1^st ‘𝐵))}, {𝑦 ∈ Q ∣ ∃𝑧(𝑧 ∈ (1^st ‘𝐴) ∧ (𝑧 +_Q 𝑦) ∈ (2^nd ‘𝐵))}⟩) = 𝐵) → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)
34	15, 30, 33	syl2anc 403	1 ⊢ (𝐴<_P 𝐵 → ∃𝑥 ∈ P (𝐴 +_P 𝑥) = 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∃wrex 2349 {crab 2352 ⟨cop 3401 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 1^st c1st 5785 2^nd c2nd 5786 Qcnq 6470 +_Q cplq 6472 Pcnp 6481 +_P cpp 6483 <_P cltp 6485

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658 df-iltp 6660

This theorem is referenced by: lteupri 6807 ltaprlem 6808 ltaprg 6809 ltmprr 6832 recexgt0sr 6950 mulgt0sr 6954

W3C validator