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Theorem recexpr 8011

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 4136	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑧 <_Q 𝑤 ↔ 𝑢 <_Q 𝑣))
2		simpr 110	. . . . . . . . 9 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑤 = 𝑣)
3	2	fveq2d 5702	. . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (_Q‘𝑤) = (_Q‘𝑣))
4	3	eleq1d 2307	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (2^nd ‘𝐴) ↔ (_Q‘𝑣) ∈ (2^nd ‘𝐴)))
5	1, 4	anbi12d 477	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ (𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
6	5	cbvexdva 1985	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ ∃𝑣(𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
7	6	cbvabv 2365	. . . 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 4144	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑤 <_Q 𝑧 ↔ 𝑣 <_Q 𝑢))
10	3	eleq1d 2307	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (1^st ‘𝐴) ↔ (_Q‘𝑣) ∈ (1^st ‘𝐴)))
11	9, 10	anbi12d 477	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ (𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
12	11	cbvexdva 1985	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ ∃𝑣(𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
13	12	cbvabv 2365	. . . 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 3909	. . 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 8005	. 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 8010	. 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 6096	. . . 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 2247	. . 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 2929	. 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 415	1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·_P 𝑥) = 1_P)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 = wceq 1402 ∃wex 1545 ∈ wcel 2209 {cab 2224 ∃wrex 2529 ⟨cop 3712 class class class wbr 4131 ‘cfv 5380 (class class class)co 6088 1^st c1st 6376 2^nd c2nd 6377 *_Qcrq 7657 <_Q cltq 7658 Pcnp 7664 1_Pc1p 7665 ·_P cmp 7667

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-po 4443 df-iso 4444 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-2o 6692 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-enq0 7797 df-nq0 7798 df-0nq0 7799 df-plq0 7800 df-mq0 7801 df-inp 7839 df-i1p 7840 df-imp 7842

This theorem is used by: ltmprr 8015 recexgt0sr 8146

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