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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 10342 | . . 3 ⊢ 1Q ∈ Q | |
2 | mulpqnq 10355 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 1Q ∈ Q) → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q))) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = ([Q]‘(𝐴 ·pQ 1Q))) |
4 | relxp 5566 | . . . . . . 7 ⊢ Rel (N × N) | |
5 | elpqn 10339 | . . . . . . 7 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N)) | |
6 | 1st2nd 7730 | . . . . . . 7 ⊢ ((Rel (N × N) ∧ 𝐴 ∈ (N × N)) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
7 | 4, 5, 6 | sylancr 589 | . . . . . 6 ⊢ (𝐴 ∈ Q → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
8 | df-1nq 10330 | . . . . . . 7 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1Q = 〈1o, 1o〉) |
10 | 7, 9 | oveq12d 7166 | . . . . 5 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉)) |
11 | xp1st 7713 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N) | |
12 | 5, 11 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N) |
13 | xp2nd 7714 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N) | |
14 | 5, 13 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N) |
15 | 1pi 10297 | . . . . . . 7 ⊢ 1o ∈ N | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝐴 ∈ Q → 1o ∈ N) |
17 | mulpipq 10354 | . . . . . 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 836 | . . . . 5 ⊢ (𝐴 ∈ Q → (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ·pQ 〈1o, 1o〉) = 〈((1st ‘𝐴) ·N 1o), ((2nd ‘𝐴) ·N 1o)〉) |
19 | mulidpi 10300 | . . . . . . . 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 10300 | . . . . . . . 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 4803 | . . . . . 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 2858 | . . . 4 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) |
26 | 25, 7 | eqtr4d 2857 | . . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·pQ 1Q) = 𝐴) |
27 | 26 | fveq2d 6667 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘(𝐴 ·pQ 1Q)) = ([Q]‘𝐴)) |
28 | nqerid 10347 | . 2 ⊢ (𝐴 ∈ Q → ([Q]‘𝐴) = 𝐴) | |
29 | 3, 27, 28 | 3eqtrd 2858 | 1 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 〈cop 4565 × cxp 5546 Rel wrel 5553 ‘cfv 6348 (class class class)co 7148 1st c1st 7679 2nd c2nd 7680 1oc1o 8087 Ncnpi 10258 ·N cmi 10260 ·pQ cmpq 10263 Qcnq 10266 1Qc1q 10267 [Q]cerq 10268 ·Q cmq 10270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-omul 8099 df-er 8281 df-ni 10286 df-mi 10288 df-lti 10289 df-mpq 10323 df-enq 10325 df-nq 10326 df-erq 10327 df-mq 10329 df-1nq 10330 |
This theorem is referenced by: recmulnq 10378 ltaddnq 10388 halfnq 10390 ltrnq 10393 addclprlem1 10430 addclprlem2 10431 mulclprlem 10433 1idpr 10443 prlem934 10447 prlem936 10461 reclem3pr 10463 |
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