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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 10966 | . . 3 ⊢ 1Q ∈ Q | |
2 | mulpqnq 10979 | . . 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 5707 | . . . . . . 7 ⊢ Rel (N × N) | |
5 | elpqn 10963 | . . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
6 | 1st2nd 8063 | . . . . . . 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 10954 | . . . . . . 7 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = 〈1o, 1o〉) |
10 | 7, 9 | oveq12d 7449 | . . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉)) |
11 | xp1st 8045 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
12 | 5, 11 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N) |
13 | xp2nd 8046 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
14 | 5, 13 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N) |
15 | 1pi 10921 | . . . . . . 7 ⊢ 1o ∈ N | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N) |
17 | mulpipq 10978 | . . . . . 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 839 | . . . . 5 ⊢ (𝐴 ∈ Q → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉) = 〈((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)〉) |
19 | mulidpi 10924 | . . . . . . . 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 10924 | . . . . . . . 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 4886 | . . . . . 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 2779 | . . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
26 | 25, 7 | eqtr4d 2778 | . . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴) |
27 | 26 | fveq2d 6911 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴)) |
28 | nqerid 10971 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) | |
29 | 3, 27, 28 | 3eqtrd 2779 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 〈cop 4637 × cxp 5687 Rel wrel 5694 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 1oc1o 8498 Ncnpi 10882 ·N cmi 10884 ·pQ cmpq 10887 Qcnq 10890 1Qc1q 10891 [Q]cerq 10892 ·Q cmq 10894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ni 10910 df-mi 10912 df-lti 10913 df-mpq 10947 df-enq 10949 df-nq 10950 df-erq 10951 df-mq 10953 df-1nq 10954 |
This theorem is referenced by: recmulnq 11002 ltaddnq 11012 halfnq 11014 ltrnq 11017 addclprlem1 11054 addclprlem2 11055 mulclprlem 11057 1idpr 11067 prlem934 11071 prlem936 11085 reclem3pr 11087 |
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