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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 10507 | . . 3 ⊢ 1Q ∈ Q | |
2 | mulpqnq 10520 | . . 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 5554 | . . . . . . 7 ⊢ Rel (N × N) | |
5 | elpqn 10504 | . . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
6 | 1st2nd 7788 | . . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
7 | 4, 5, 6 | sylancr 590 | . . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
8 | df-1nq 10495 | . . . . . . 7 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = 〈1o, 1o〉) |
10 | 7, 9 | oveq12d 7209 | . . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉)) |
11 | xp1st 7771 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
12 | 5, 11 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N) |
13 | xp2nd 7772 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
14 | 5, 13 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N) |
15 | 1pi 10462 | . . . . . . 7 ⊢ 1o ∈ N | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N) |
17 | mulpipq 10519 | . . . . . 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 10465 | . . . . . . . 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 10465 | . . . . . . . 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 4778 | . . . . . 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 2775 | . . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
26 | 25, 7 | eqtr4d 2774 | . . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴) |
27 | 26 | fveq2d 6699 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴)) |
28 | nqerid 10512 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) | |
29 | 3, 27, 28 | 3eqtrd 2775 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 〈cop 4533 × cxp 5534 Rel wrel 5541 ‘cfv 6358 (class class class)co 7191 1st c1st 7737 2nd c2nd 7738 1oc1o 8173 Ncnpi 10423 ·N cmi 10425 ·pQ cmpq 10428 Qcnq 10431 1Qc1q 10432 [Q]cerq 10433 ·Q cmq 10435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-omul 8185 df-er 8369 df-ni 10451 df-mi 10453 df-lti 10454 df-mpq 10488 df-enq 10490 df-nq 10491 df-erq 10492 df-mq 10494 df-1nq 10495 |
This theorem is referenced by: recmulnq 10543 ltaddnq 10553 halfnq 10555 ltrnq 10558 addclprlem1 10595 addclprlem2 10596 mulclprlem 10598 1idpr 10608 prlem934 10612 prlem936 10626 reclem3pr 10628 |
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