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Theorem recexpr 7347

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 3880	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑧 <_Q 𝑤 ↔ 𝑢 <_Q 𝑣))
2		simpr 109	. . . . . . . . 9 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑤 = 𝑣)
3	2	fveq2d 5357	. . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (_Q‘𝑤) = (_Q‘𝑣))
4	3	eleq1d 2168	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (2^nd ‘𝐴) ↔ (_Q‘𝑣) ∈ (2^nd ‘𝐴)))
5	1, 4	anbi12d 460	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ (𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
6	5	cbvexdva 1864	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ ∃𝑣(𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
7	6	cbvabv 2223	. . . 4 ⊢ {𝑧 ∣ ∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))}
8		simpl 108	. . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑧 = 𝑢)
9	2, 8	breq12d 3888	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑤 <_Q 𝑧 ↔ 𝑣 <_Q 𝑢))
10	3	eleq1d 2168	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (1^st ‘𝐴) ↔ (_Q‘𝑣) ∈ (1^st ‘𝐴)))
11	9, 10	anbi12d 460	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ (𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
12	11	cbvexdva 1864	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ ∃𝑣(𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
13	12	cbvabv 2223	. . . 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 3657	. . 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 7341	. 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 7346	. 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 5714	. . . 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 2108	. . 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 2744	. 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 406	1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·_P 𝑥) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∃wex 1436 ∈ wcel 1448 {cab 2086 ∃wrex 2376 ⟨cop 3477 class class class wbr 3875 ‘cfv 5059 (class class class)co 5706 1^st c1st 5967 2^nd c2nd 5968 *_Qcrq 6993 <_Q cltq 6994 Pcnp 7000 1_Pc1p 7001 ·_P cmp 7003

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440

This theorem depends on definitions: df-bi 116 df-dc 787 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-eprel 4149 df-id 4153 df-po 4156 df-iso 4157 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-1o 6243 df-2o 6244 df-oadd 6247 df-omul 6248 df-er 6359 df-ec 6361 df-qs 6365 df-ni 7013 df-pli 7014 df-mi 7015 df-lti 7016 df-plpq 7053 df-mpq 7054 df-enq 7056 df-nqqs 7057 df-plqqs 7058 df-mqqs 7059 df-1nqqs 7060 df-rq 7061 df-ltnqqs 7062 df-enq0 7133 df-nq0 7134 df-0nq0 7135 df-plq0 7136 df-mq0 7137 df-inp 7175 df-i1p 7176 df-imp 7178

This theorem is referenced by: ltmprr 7351 recexgt0sr 7469

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