Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9629 | . . 3 ⊢ 1Q ∈ Q | |
2 | mulpqnq 9642 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q))) | |
3 | 1, 2 | mpan2 703 | . 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q))) |
4 | relxp 5150 | . . . . . . 7 ⊢ Rel (N × N) | |
5 | elpqn 9626 | . . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
6 | 1st2nd 7105 | . . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
7 | 4, 5, 6 | sylancr 694 | . . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
8 | df-1nq 9617 | . . . . . . 7 ⊢ 1Q = 〈1𝑜, 1𝑜〉 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = 〈1𝑜, 1𝑜〉) |
10 | 7, 9 | oveq12d 6567 | . . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1𝑜, 1𝑜〉)) |
11 | xp1st 7089 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
12 | 5, 11 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N) |
13 | xp2nd 7090 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
14 | 5, 13 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N) |
15 | 1pi 9584 | . . . . . . 7 ⊢ 1𝑜 ∈ N | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1𝑜 ∈ N) |
17 | mulpipq 9641 | . . . . . 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 1319 | . . . . 5 ⊢ (𝐴 ∈ Q → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1𝑜, 1𝑜〉) = 〈((1st ‘𝐴) ·N 1𝑜), ((2nd ‘𝐴) ·N 1𝑜)〉) |
19 | mulidpi 9587 | . . . . . . . 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 9587 | . . . . . . . 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 4348 | . . . . . 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 2648 | . . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
26 | 25, 7 | eqtr4d 2647 | . . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴) |
27 | 26 | fveq2d 6107 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴)) |
28 | nqerid 9634 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) | |
29 | 3, 27, 28 | 3eqtrd 2648 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 × cxp 5036 Rel wrel 5043 ‘cfv 5804 (class class class)co 6549 1st c1st 7057 2nd c2nd 7058 1𝑜c1o 7440 Ncnpi 9545 ·N cmi 9547 ·pQ cmpq 9550 Qcnq 9553 1Qc1q 9554 [Q]cerq 9555 ·Q cmq 9557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ni 9573 df-mi 9575 df-lti 9576 df-mpq 9610 df-enq 9612 df-nq 9613 df-erq 9614 df-mq 9616 df-1nq 9617 |
This theorem is referenced by: recmulnq 9665 ltaddnq 9675 halfnq 9677 ltrnq 9680 addclprlem1 9717 addclprlem2 9718 mulclprlem 9720 1idpr 9730 prlem934 9734 prlem936 9748 reclem3pr 9750 |
Copyright terms: Public domain | W3C validator |