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| Mirrors > Home > MPE Home > Th. List > mulidnq | Structured version Visualization version GIF version | ||
| Description: Multiplication identity element for positive fractions. (Contributed by NM, 3-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| mulidnq | ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 1nq 10969 | . . 3 ⊢ 1Q ∈ Q | |
| 2 | mulpqnq 10982 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q))) | |
| 3 | 1, 2 | mpan2 691 | . 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q))) | 
| 4 | relxp 5702 | . . . . . . 7 ⊢ Rel (N × N) | |
| 5 | elpqn 10966 | . . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
| 6 | 1st2nd 8065 | . . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
| 7 | 4, 5, 6 | sylancr 587 | . . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | 
| 8 | df-1nq 10957 | . . . . . . 7 ⊢ 1Q = 〈1o, 1o〉 | |
| 9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = 〈1o, 1o〉) | 
| 10 | 7, 9 | oveq12d 7450 | . . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉)) | 
| 11 | xp1st 8047 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
| 12 | 5, 11 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N) | 
| 13 | xp2nd 8048 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
| 14 | 5, 13 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N) | 
| 15 | 1pi 10924 | . . . . . . 7 ⊢ 1o ∈ N | |
| 16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N) | 
| 17 | mulpipq 10981 | . . . . . 6 ⊢ ((((1st ‘𝐴) ∈ N ∧ (2nd ‘𝐴) ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉) = 〈((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)〉) | |
| 18 | 12, 14, 16, 16, 17 | syl22anc 838 | . . . . 5 ⊢ (𝐴 ∈ Q → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉) = 〈((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)〉) | 
| 19 | mulidpi 10927 | . . . . . . . 8 ⊢ ((1st ‘𝐴) ∈ N → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴)) | |
| 20 | 11, 19 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((1st ‘𝐴) ·N 1o) = (1st ‘𝐴)) | 
| 21 | mulidpi 10927 | . . . . . . . 8 ⊢ ((2nd ‘𝐴) ∈ N → ((2nd ‘𝐴) ·N 1o) = (2nd ‘𝐴)) | |
| 22 | 13, 21 | syl 17 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → ((2nd ‘𝐴) ·N 1o) = (2nd ‘𝐴)) | 
| 23 | 20, 22 | opeq12d 4880 | . . . . . 6 ⊢ (𝐴 ∈ (N × N) → 〈((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)〉 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | 
| 24 | 5, 23 | syl 17 | . . . . 5 ⊢ (𝐴 ∈ Q → 〈((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)〉 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | 
| 25 | 10, 18, 24 | 3eqtrd 2780 | . . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | 
| 26 | 25, 7 | eqtr4d 2779 | . . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴) | 
| 27 | 26 | fveq2d 6909 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴)) | 
| 28 | nqerid 10974 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) | |
| 29 | 3, 27, 28 | 3eqtrd 2780 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 〈cop 4631 × cxp 5682 Rel wrel 5689 ‘cfv 6560 (class class class)co 7432 1st c1st 8013 2nd c2nd 8014 1oc1o 8500 Ncnpi 10885 ·N cmi 10887 ·pQ cmpq 10890 Qcnq 10893 1Qc1q 10894 [Q]cerq 10895 ·Q cmq 10897 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-omul 8512 df-er 8746 df-ni 10913 df-mi 10915 df-lti 10916 df-mpq 10950 df-enq 10952 df-nq 10953 df-erq 10954 df-mq 10956 df-1nq 10957 | 
| This theorem is referenced by: recmulnq 11005 ltaddnq 11015 halfnq 11017 ltrnq 11020 addclprlem1 11057 addclprlem2 11058 mulclprlem 11060 1idpr 11070 prlem934 11074 prlem936 11088 reclem3pr 11090 | 
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