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Mirrors > Home > ILE Home > Th. List > recmulnqg | GIF version |
Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by Jim Kingdon, 19-Sep-2019.) |
Ref | Expression |
---|---|
recmulnqg | ⊢ ((A ∈ Q ∧ B ∈ Q) → ((*Q‘A) = B ↔ (A ·Q B) = 1Q)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5462 | . . . . 5 ⊢ (x = A → (x ·Q y) = (A ·Q y)) | |
2 | 1 | eqeq1d 2045 | . . . 4 ⊢ (x = A → ((x ·Q y) = 1Q ↔ (A ·Q y) = 1Q)) |
3 | 2 | anbi2d 437 | . . 3 ⊢ (x = A → ((y ∈ Q ∧ (x ·Q y) = 1Q) ↔ (y ∈ Q ∧ (A ·Q y) = 1Q))) |
4 | eleq1 2097 | . . . 4 ⊢ (y = B → (y ∈ Q ↔ B ∈ Q)) | |
5 | oveq2 5463 | . . . . 5 ⊢ (y = B → (A ·Q y) = (A ·Q B)) | |
6 | 5 | eqeq1d 2045 | . . . 4 ⊢ (y = B → ((A ·Q y) = 1Q ↔ (A ·Q B) = 1Q)) |
7 | 4, 6 | anbi12d 442 | . . 3 ⊢ (y = B → ((y ∈ Q ∧ (A ·Q y) = 1Q) ↔ (B ∈ Q ∧ (A ·Q B) = 1Q))) |
8 | recexnq 6374 | . . . 4 ⊢ (x ∈ Q → ∃y(y ∈ Q ∧ (x ·Q y) = 1Q)) | |
9 | 1nq 6350 | . . . . 5 ⊢ 1Q ∈ Q | |
10 | mulcomnqg 6367 | . . . . 5 ⊢ ((z ∈ Q ∧ w ∈ Q) → (z ·Q w) = (w ·Q z)) | |
11 | mulassnqg 6368 | . . . . 5 ⊢ ((z ∈ Q ∧ w ∈ Q ∧ v ∈ Q) → ((z ·Q w) ·Q v) = (z ·Q (w ·Q v))) | |
12 | mulidnq 6373 | . . . . 5 ⊢ (z ∈ Q → (z ·Q 1Q) = z) | |
13 | 9, 10, 11, 12 | caovimo 5636 | . . . 4 ⊢ (x ∈ Q → ∃*y(y ∈ Q ∧ (x ·Q y) = 1Q)) |
14 | eu5 1944 | . . . 4 ⊢ (∃!y(y ∈ Q ∧ (x ·Q y) = 1Q) ↔ (∃y(y ∈ Q ∧ (x ·Q y) = 1Q) ∧ ∃*y(y ∈ Q ∧ (x ·Q y) = 1Q))) | |
15 | 8, 13, 14 | sylanbrc 394 | . . 3 ⊢ (x ∈ Q → ∃!y(y ∈ Q ∧ (x ·Q y) = 1Q)) |
16 | df-rq 6336 | . . . 4 ⊢ *Q = {〈x, y〉 ∣ (x ∈ Q ∧ y ∈ Q ∧ (x ·Q y) = 1Q)} | |
17 | 3anass 888 | . . . . 5 ⊢ ((x ∈ Q ∧ y ∈ Q ∧ (x ·Q y) = 1Q) ↔ (x ∈ Q ∧ (y ∈ Q ∧ (x ·Q y) = 1Q))) | |
18 | 17 | opabbii 3815 | . . . 4 ⊢ {〈x, y〉 ∣ (x ∈ Q ∧ y ∈ Q ∧ (x ·Q y) = 1Q)} = {〈x, y〉 ∣ (x ∈ Q ∧ (y ∈ Q ∧ (x ·Q y) = 1Q))} |
19 | 16, 18 | eqtri 2057 | . . 3 ⊢ *Q = {〈x, y〉 ∣ (x ∈ Q ∧ (y ∈ Q ∧ (x ·Q y) = 1Q))} |
20 | 3, 7, 15, 19 | fvopab3g 5188 | . 2 ⊢ ((A ∈ Q ∧ B ∈ Q) → ((*Q‘A) = B ↔ (B ∈ Q ∧ (A ·Q B) = 1Q))) |
21 | ibar 285 | . . 3 ⊢ (B ∈ Q → ((A ·Q B) = 1Q ↔ (B ∈ Q ∧ (A ·Q B) = 1Q))) | |
22 | 21 | adantl 262 | . 2 ⊢ ((A ∈ Q ∧ B ∈ Q) → ((A ·Q B) = 1Q ↔ (B ∈ Q ∧ (A ·Q B) = 1Q))) |
23 | 20, 22 | bitr4d 180 | 1 ⊢ ((A ∈ Q ∧ B ∈ Q) → ((*Q‘A) = B ↔ (A ·Q B) = 1Q)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃!weu 1897 ∃*wmo 1898 {copab 3808 ‘cfv 4845 (class class class)co 5455 Qcnq 6264 1Qc1q 6265 ·Q cmq 6267 *Qcrq 6268 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-mi 6290 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 |
This theorem is referenced by: recclnq 6376 recidnq 6377 recrecnq 6378 recexprlem1ssl 6605 recexprlem1ssu 6606 |
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