Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mulidnq | GIF version |
Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) |
Ref | Expression |
---|---|
mulidnq | ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nqqs 7303 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | oveq1 5858 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = (𝐴 ·Q 1Q)) | |
3 | id 19 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → [〈𝑥, 𝑦〉] ~Q = 𝐴) | |
4 | 2, 3 | eqeq12d 2185 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → (([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴)) |
5 | df-1nqqs 7306 | . . . . 5 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
6 | 5 | oveq2i 5862 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = ([〈𝑥, 𝑦〉] ~Q ·Q [〈1o, 1o〉] ~Q ) |
7 | 1pi 7270 | . . . . 5 ⊢ 1o ∈ N | |
8 | mulpipqqs 7328 | . . . . 5 ⊢ (((𝑥 ∈ N ∧ 𝑦 ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈1o, 1o〉] ~Q ) = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) | |
9 | 7, 7, 8 | mpanr12 437 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈1o, 1o〉] ~Q ) = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
10 | 6, 9 | eqtrid 2215 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
11 | mulcompig 7286 | . . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o)) | |
12 | 7, 11 | mpan 422 | . . . . . 6 ⊢ (𝑥 ∈ N → (1o ·N 𝑥) = (𝑥 ·N 1o)) |
13 | 12 | adantr 274 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o)) |
14 | mulcompig 7286 | . . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o)) | |
15 | 7, 14 | mpan 422 | . . . . . 6 ⊢ (𝑦 ∈ N → (1o ·N 𝑦) = (𝑦 ·N 1o)) |
16 | 15 | adantl 275 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o)) |
17 | 13, 16 | opeq12d 3771 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → 〈(1o ·N 𝑥), (1o ·N 𝑦)〉 = 〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉) |
18 | 17 | eceq1d 6547 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
19 | mulcanenqec 7341 | . . . 4 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) | |
20 | 7, 19 | mp3an1 1319 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) |
21 | 10, 18, 20 | 3eqtr2d 2209 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ) |
22 | 1, 4, 21 | ecoptocl 6598 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 〈cop 3584 (class class class)co 5851 1oc1o 6386 [cec 6509 Ncnpi 7227 ·N cmi 7229 ~Q ceq 7234 Qcnq 7235 1Qc1q 7236 ·Q cmq 7238 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-irdg 6347 df-1o 6393 df-oadd 6397 df-omul 6398 df-er 6511 df-ec 6513 df-qs 6517 df-ni 7259 df-mi 7261 df-mpq 7300 df-enq 7302 df-nqqs 7303 df-mqqs 7305 df-1nqqs 7306 |
This theorem is referenced by: recmulnqg 7346 rec1nq 7350 ltaddnq 7362 halfnqq 7365 prarloclemarch 7373 ltrnqg 7375 addnqprllem 7482 addnqprulem 7483 addnqprl 7484 addnqpru 7485 appdivnq 7518 prmuloc2 7522 mulnqprl 7523 mulnqpru 7524 1idprl 7545 1idpru 7546 recexprlem1ssl 7588 recexprlem1ssu 7589 |
Copyright terms: Public domain | W3C validator |