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Theorem recexpr 7848

Description: The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)

**Assertion**
Ref	Expression
recexpr	⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·_P 𝑥) = 1_P)

Distinct variable group: 𝑥,𝐴

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

Step	Hyp	Ref	Expression
1		breq12 4091	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑧 <_Q 𝑤 ↔ 𝑢 <_Q 𝑣))
2		simpr 110	. . . . . . . . 9 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑤 = 𝑣)
3	2	fveq2d 5639	. . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (_Q‘𝑤) = (_Q‘𝑣))
4	3	eleq1d 2298	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (2^nd ‘𝐴) ↔ (_Q‘𝑣) ∈ (2^nd ‘𝐴)))
5	1, 4	anbi12d 473	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ (𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
6	5	cbvexdva 1976	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ ∃𝑣(𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
7	6	cbvabv 2354	. . . 4 ⊢ {𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))}
8		simpl 109	. . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑧 = 𝑢)
9	2, 8	breq12d 4099	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑤 <_Q 𝑧 ↔ 𝑣 <_Q 𝑢))
10	3	eleq1d 2298	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (1^st ‘𝐴) ↔ (_Q‘𝑣) ∈ (1^st ‘𝐴)))
11	9, 10	anbi12d 473	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ (𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
12	11	cbvexdva 1976	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ ∃𝑣(𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
13	12	cbvabv 2354	. . . 4 ⊢ {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))}
14	7, 13	opeq12i 3865	. . 3 ⊢ ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩ = ⟨{𝑢 ∣ ∃𝑣(𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))}, {𝑢 ∣ ∃𝑣(𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))}⟩
15	14	recexprlempr 7842	. 2 ⊢ (𝐴 ∈ P → ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩ ∈ P)
16	14	recexprlemex 7847	. 2 ⊢ (𝐴 ∈ P → (𝐴 ·_P ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩) = 1_P)
17		oveq2 6021	. . . 4 ⊢ (𝑥 = ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩ → (𝐴 ·_P 𝑥) = (𝐴 ·_P ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩))
18	17	eqeq1d 2238	. . 3 ⊢ (𝑥 = ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩ → ((𝐴 ·_P 𝑥) = 1_P ↔ (𝐴 ·_P ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩) = 1_P))
19	18	rspcev 2908	. 2 ⊢ ((⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩ ∈ P ∧ (𝐴 ·_P ⟨{𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴))}⟩) = 1_P) → ∃𝑥 ∈ P (𝐴 ·_P 𝑥) = 1_P)
20	15, 16, 19	syl2anc 411	1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·_P 𝑥) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∃wex 1538 ∈ wcel 2200 {cab 2215 ∃wrex 2509 ⟨cop 3670 class class class wbr 4086 ‘cfv 5324 (class class class)co 6013 1^st c1st 6296 2^nd c2nd 6297 *_Qcrq 7494 <_Q cltq 7495 Pcnp 7501 1_Pc1p 7502 ·_P cmp 7504

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-eprel 4384 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-irdg 6531 df-1o 6577 df-2o 6578 df-oadd 6581 df-omul 6582 df-er 6697 df-ec 6699 df-qs 6703 df-ni 7514 df-pli 7515 df-mi 7516 df-lti 7517 df-plpq 7554 df-mpq 7555 df-enq 7557 df-nqqs 7558 df-plqqs 7559 df-mqqs 7560 df-1nqqs 7561 df-rq 7562 df-ltnqqs 7563 df-enq0 7634 df-nq0 7635 df-0nq0 7636 df-plq0 7637 df-mq0 7638 df-inp 7676 df-i1p 7677 df-imp 7679

This theorem is referenced by: ltmprr 7852 recexgt0sr 7983

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