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Mirrors > Home > MPE Home > Th. List > 1nqenq | Structured version Visualization version GIF version |
Description: The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1nqenq | ⊢ (𝐴 ∈ N → 1Q ~Q 〈𝐴, 𝐴〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enqer 10964 | . . 3 ⊢ ~Q Er (N × N) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ N → ~Q Er (N × N)) |
3 | mulidpi 10929 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
4 | 3, 3 | opeq12d 4887 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 = 〈𝐴, 𝐴〉) |
5 | 1pi 10926 | . . . . 5 ⊢ 1o ∈ N | |
6 | mulcanenq 11003 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N ∧ 1o ∈ N) → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) | |
7 | 5, 5, 6 | mp3an23 1450 | . . . 4 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) |
8 | df-1nq 10959 | . . . 4 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 7, 8 | breqtrrdi 5195 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 1Q) |
10 | 4, 9 | eqbrtrrd 5177 | . 2 ⊢ (𝐴 ∈ N → 〈𝐴, 𝐴〉 ~Q 1Q) |
11 | 2, 10 | ersym 8746 | 1 ⊢ (𝐴 ∈ N → 1Q ~Q 〈𝐴, 𝐴〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 〈cop 4639 class class class wbr 5153 × cxp 5680 (class class class)co 7424 1oc1o 8489 Er wer 8731 Ncnpi 10887 ·N cmi 10889 ~Q ceq 10894 1Qc1q 10896 |
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-1o 8496 df-oadd 8500 df-omul 8501 df-er 8734 df-ni 10915 df-mi 10917 df-enq 10954 df-1nq 10959 |
This theorem is referenced by: recmulnq 11007 |
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