![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > enqer | Structured version Visualization version GIF version |
Description: The equivalence relation for positive fractions is an equivalence relation. Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
enqer | ⊢ ~Q Er (N × N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-enq 10954 | . 2 ⊢ ~Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} | |
2 | mulcompi 10939 | . 2 ⊢ (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥) | |
3 | mulclpi 10936 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → (𝑥 ·N 𝑦) ∈ N) | |
4 | mulasspi 10940 | . 2 ⊢ ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧)) | |
5 | mulcanpi 10943 | . . 3 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) ↔ 𝑦 = 𝑧)) | |
6 | 5 | biimpd 228 | . 2 ⊢ ((𝑥 ∈ N ∧ 𝑦 ∈ N) → ((𝑥 ·N 𝑦) = (𝑥 ·N 𝑧) → 𝑦 = 𝑧)) |
7 | 1, 2, 3, 4, 6 | ecopover 8850 | 1 ⊢ ~Q Er (N × N) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∈ wcel 2099 × cxp 5680 (class class class)co 7424 Er wer 8731 Ncnpi 10887 ·N cmi 10889 ~Q ceq 10894 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-oadd 8500 df-omul 8501 df-er 8734 df-ni 10915 df-mi 10917 df-enq 10954 |
This theorem is referenced by: nqereu 10972 nqerf 10973 nqerid 10976 enqeq 10977 nqereq 10978 adderpq 10999 mulerpq 11000 1nqenq 11005 |
Copyright terms: Public domain | W3C validator |