![]() |
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 6968 | . 2 ⊢ Q = ((N × N) / ~Q ) | |
2 | oveq1 5673 | . . 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 2103 | . 2 ⊢ ([〈𝑥, 𝑦〉] ~Q = 𝐴 → (([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ↔ (𝐴 ·Q 1Q) = 𝐴)) |
5 | df-1nqqs 6971 | . . . . 5 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
6 | 5 | oveq2i 5677 | . . . 4 ⊢ ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = ([〈𝑥, 𝑦〉] ~Q ·Q [〈1o, 1o〉] ~Q ) |
7 | 1pi 6935 | . . . . 5 ⊢ 1o ∈ N | |
8 | mulpipqqs 6993 | . . . . 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 431 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q [〈1o, 1o〉] ~Q ) = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
10 | 6, 9 | syl5eq 2133 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
11 | mulcompig 6951 | . . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o)) | |
12 | 7, 11 | mpan 416 | . . . . . 6 ⊢ (𝑥 ∈ N → (1o ·N 𝑥) = (𝑥 ·N 1o)) |
13 | 12 | adantr 271 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑥) = (𝑥 ·N 1o)) |
14 | mulcompig 6951 | . . . . . . 7 ⊢ ((1o ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o)) | |
15 | 7, 14 | mpan 416 | . . . . . 6 ⊢ (𝑦 ∈ N → (1o ·N 𝑦) = (𝑦 ·N 1o)) |
16 | 15 | adantl 272 | . . . . 5 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (1o ·N 𝑦) = (𝑦 ·N 1o)) |
17 | 13, 16 | opeq12d 3636 | . . . 4 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → 〈(1o ·N 𝑥), (1o ·N 𝑦)〉 = 〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉) |
18 | 17 | eceq1d 6342 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈(𝑥 ·N 1o), (𝑦 ·N 1o)〉] ~Q ) |
19 | mulcanenqec 7006 | . . . 4 ⊢ ((1o ∈ N ∧ 𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) | |
20 | 7, 19 | mp3an1 1261 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → [〈(1o ·N 𝑥), (1o ·N 𝑦)〉] ~Q = [〈𝑥, 𝑦〉] ~Q ) |
21 | 10, 18, 20 | 3eqtr2d 2127 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ([〈𝑥, 𝑦〉] ~Q ·Q 1Q) = [〈𝑥, 𝑦〉] ~Q ) |
22 | 1, 4, 21 | ecoptocl 6393 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 〈cop 3453 (class class class)co 5666 1oc1o 6188 [cec 6304 Ncnpi 6892 ·N cmi 6894 ~Q ceq 6899 Qcnq 6900 1Qc1q 6901 ·Q cmq 6903 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6924 df-mi 6926 df-mpq 6965 df-enq 6967 df-nqqs 6968 df-mqqs 6970 df-1nqqs 6971 |
This theorem is referenced by: recmulnqg 7011 rec1nq 7015 ltaddnq 7027 halfnqq 7030 prarloclemarch 7038 ltrnqg 7040 addnqprllem 7147 addnqprulem 7148 addnqprl 7149 addnqpru 7150 appdivnq 7183 prmuloc2 7187 mulnqprl 7188 mulnqpru 7189 1idprl 7210 1idpru 7211 recexprlem1ssl 7253 recexprlem1ssu 7254 |
Copyright terms: Public domain | W3C validator |