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Mirrors > Home > MPE Home > Th. List > recexsr | Structured version Visualization version GIF version |
Description: The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
Ref | Expression |
---|---|
recexsr | ⊢ ((𝐴 ∈ R ∧ 𝐴 ≠ 0R) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqgt0sr 10111 | . 2 ⊢ ((𝐴 ∈ R ∧ 𝐴 ≠ 0R) → 0R <R (𝐴 ·R 𝐴)) | |
2 | recexsrlem 10108 | . . . 4 ⊢ (0R <R (𝐴 ·R 𝐴) → ∃𝑦 ∈ R ((𝐴 ·R 𝐴) ·R 𝑦) = 1R) | |
3 | mulclsr 10089 | . . . . . . 7 ⊢ ((𝐴 ∈ R ∧ 𝑦 ∈ R) → (𝐴 ·R 𝑦) ∈ R) | |
4 | mulasssr 10095 | . . . . . . . . 9 ⊢ ((𝐴 ·R 𝐴) ·R 𝑦) = (𝐴 ·R (𝐴 ·R 𝑦)) | |
5 | 4 | eqeq1i 2757 | . . . . . . . 8 ⊢ (((𝐴 ·R 𝐴) ·R 𝑦) = 1R ↔ (𝐴 ·R (𝐴 ·R 𝑦)) = 1R) |
6 | oveq2 6813 | . . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ·R 𝑦) → (𝐴 ·R 𝑥) = (𝐴 ·R (𝐴 ·R 𝑦))) | |
7 | 6 | eqeq1d 2754 | . . . . . . . . 9 ⊢ (𝑥 = (𝐴 ·R 𝑦) → ((𝐴 ·R 𝑥) = 1R ↔ (𝐴 ·R (𝐴 ·R 𝑦)) = 1R)) |
8 | 7 | rspcev 3441 | . . . . . . . 8 ⊢ (((𝐴 ·R 𝑦) ∈ R ∧ (𝐴 ·R (𝐴 ·R 𝑦)) = 1R) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) |
9 | 5, 8 | sylan2b 493 | . . . . . . 7 ⊢ (((𝐴 ·R 𝑦) ∈ R ∧ ((𝐴 ·R 𝐴) ·R 𝑦) = 1R) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) |
10 | 3, 9 | sylan 489 | . . . . . 6 ⊢ (((𝐴 ∈ R ∧ 𝑦 ∈ R) ∧ ((𝐴 ·R 𝐴) ·R 𝑦) = 1R) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) |
11 | 10 | exp31 631 | . . . . 5 ⊢ (𝐴 ∈ R → (𝑦 ∈ R → (((𝐴 ·R 𝐴) ·R 𝑦) = 1R → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R))) |
12 | 11 | rexlimdv 3160 | . . . 4 ⊢ (𝐴 ∈ R → (∃𝑦 ∈ R ((𝐴 ·R 𝐴) ·R 𝑦) = 1R → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R)) |
13 | 2, 12 | syl5 34 | . . 3 ⊢ (𝐴 ∈ R → (0R <R (𝐴 ·R 𝐴) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R)) |
14 | 13 | imp 444 | . 2 ⊢ ((𝐴 ∈ R ∧ 0R <R (𝐴 ·R 𝐴)) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) |
15 | 1, 14 | syldan 488 | 1 ⊢ ((𝐴 ∈ R ∧ 𝐴 ≠ 0R) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 ∃wrex 3043 class class class wbr 4796 (class class class)co 6805 Rcnr 9871 0Rc0r 9872 1Rc1r 9873 ·R cmr 9876 <R cltr 9877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-inf2 8703 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-omul 7726 df-er 7903 df-ec 7905 df-qs 7909 df-ni 9878 df-pli 9879 df-mi 9880 df-lti 9881 df-plpq 9914 df-mpq 9915 df-ltpq 9916 df-enq 9917 df-nq 9918 df-erq 9919 df-plq 9920 df-mq 9921 df-1nq 9922 df-rq 9923 df-ltnq 9924 df-np 9987 df-1p 9988 df-plp 9989 df-mp 9990 df-ltp 9991 df-enr 10061 df-nr 10062 df-plr 10063 df-mr 10064 df-ltr 10065 df-0r 10066 df-1r 10067 df-m1r 10068 |
This theorem is referenced by: axrrecex 10168 |
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