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Mirrors > Home > ILE Home > Th. List > rec1nq | GIF version |
Description: Reciprocal of positive fraction one. (Contributed by Jim Kingdon, 29-Dec-2019.) |
Ref | Expression |
---|---|
rec1nq | ⊢ (*Q‘1Q) = 1Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nq 7286 | . . . 4 ⊢ 1Q ∈ Q | |
2 | recclnq 7312 | . . . 4 ⊢ (1Q ∈ Q → (*Q‘1Q) ∈ Q) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (*Q‘1Q) ∈ Q |
4 | mulcomnqg 7303 | . . 3 ⊢ (((*Q‘1Q) ∈ Q ∧ 1Q ∈ Q) → ((*Q‘1Q) ·Q 1Q) = (1Q ·Q (*Q‘1Q))) | |
5 | 3, 1, 4 | mp2an 423 | . 2 ⊢ ((*Q‘1Q) ·Q 1Q) = (1Q ·Q (*Q‘1Q)) |
6 | mulidnq 7309 | . . 3 ⊢ ((*Q‘1Q) ∈ Q → ((*Q‘1Q) ·Q 1Q) = (*Q‘1Q)) | |
7 | 1, 2, 6 | mp2b 8 | . 2 ⊢ ((*Q‘1Q) ·Q 1Q) = (*Q‘1Q) |
8 | recidnq 7313 | . . 3 ⊢ (1Q ∈ Q → (1Q ·Q (*Q‘1Q)) = 1Q) | |
9 | 1, 8 | ax-mp 5 | . 2 ⊢ (1Q ·Q (*Q‘1Q)) = 1Q |
10 | 5, 7, 9 | 3eqtr3i 2186 | 1 ⊢ (*Q‘1Q) = 1Q |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∈ wcel 2128 ‘cfv 5170 (class class class)co 5824 Qcnq 7200 1Qc1q 7201 ·Q cmq 7203 *Qcrq 7204 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-mi 7226 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 |
This theorem is referenced by: recexprlem1ssl 7553 caucvgprlemm 7588 caucvgprprlemmu 7615 caucvgsr 7722 |
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