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Theorem recexpr 7579

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 3987	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑧 <_Q 𝑤 ↔ 𝑢 <_Q 𝑣))
2		simpr 109	. . . . . . . . 9 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑤 = 𝑣)
3	2	fveq2d 5490	. . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (_Q‘𝑤) = (_Q‘𝑣))
4	3	eleq1d 2235	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (2^nd ‘𝐴) ↔ (_Q‘𝑣) ∈ (2^nd ‘𝐴)))
5	1, 4	anbi12d 465	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ (𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
6	5	cbvexdva 1917	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑧 <_Q 𝑤 ∧ (_Q‘𝑤) ∈ (2^nd ‘𝐴)) ↔ ∃𝑣(𝑢 <_Q 𝑣 ∧ (_Q‘𝑣) ∈ (2^nd ‘𝐴))))
7	6	cbvabv 2291	. . . 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 3995	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑤 <_Q 𝑧 ↔ 𝑣 <_Q 𝑢))
10	3	eleq1d 2235	. . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((_Q‘𝑤) ∈ (1^st ‘𝐴) ↔ (_Q‘𝑣) ∈ (1^st ‘𝐴)))
11	9, 10	anbi12d 465	. . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ (𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
12	11	cbvexdva 1917	. . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑤 <_Q 𝑧 ∧ (_Q‘𝑤) ∈ (1^st ‘𝐴)) ↔ ∃𝑣(𝑣 <_Q 𝑢 ∧ (_Q‘𝑣) ∈ (1^st ‘𝐴))))
13	12	cbvabv 2291	. . . 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 3763	. . 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 7573	. 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 7578	. 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 5850	. . . 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 2174	. . 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 2830	. 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 409	1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·_P 𝑥) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1343 ∃wex 1480 ∈ wcel 2136 {cab 2151 ∃wrex 2445 ⟨cop 3579 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 1^st c1st 6106 2^nd c2nd 6107 *_Qcrq 7225 <_Q cltq 7226 Pcnp 7232 1_Pc1p 7233 ·_P cmp 7235

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-eprel 4267 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-1o 6384 df-2o 6385 df-oadd 6388 df-omul 6389 df-er 6501 df-ec 6503 df-qs 6507 df-ni 7245 df-pli 7246 df-mi 7247 df-lti 7248 df-plpq 7285 df-mpq 7286 df-enq 7288 df-nqqs 7289 df-plqqs 7290 df-mqqs 7291 df-1nqqs 7292 df-rq 7293 df-ltnqqs 7294 df-enq0 7365 df-nq0 7366 df-0nq0 7367 df-plq0 7368 df-mq0 7369 df-inp 7407 df-i1p 7408 df-imp 7410

This theorem is referenced by: ltmprr 7583 recexgt0sr 7714

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