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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 10532 | . . 3 ⊢ ~Q Er (N × N) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ N → ~Q Er (N × N)) |
3 | mulidpi 10497 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
4 | 3, 3 | opeq12d 4789 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 = 〈𝐴, 𝐴〉) |
5 | 1pi 10494 | . . . . 5 ⊢ 1o ∈ N | |
6 | mulcanenq 10571 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N ∧ 1o ∈ N) → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) | |
7 | 5, 5, 6 | mp3an23 1455 | . . . 4 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) |
8 | df-1nq 10527 | . . . 4 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 7, 8 | breqtrrdi 5092 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 1Q) |
10 | 4, 9 | eqbrtrrd 5074 | . 2 ⊢ (𝐴 ∈ N → 〈𝐴, 𝐴〉 ~Q 1Q) |
11 | 2, 10 | ersym 8400 | 1 ⊢ (𝐴 ∈ N → 1Q ~Q 〈𝐴, 𝐴〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 〈cop 4544 class class class wbr 5050 × cxp 5546 (class class class)co 7210 1oc1o 8192 Er wer 8385 Ncnpi 10455 ·N cmi 10457 ~Q ceq 10462 1Qc1q 10464 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5189 ax-nul 5196 ax-pr 5319 ax-un 7520 |
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 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-pss 3882 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-tp 4543 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-tr 5159 df-id 5452 df-eprel 5457 df-po 5465 df-so 5466 df-fr 5506 df-we 5508 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 6157 df-ord 6213 df-on 6214 df-lim 6215 df-suc 6216 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-ov 7213 df-oprab 7214 df-mpo 7215 df-om 7642 df-1st 7758 df-2nd 7759 df-wrecs 8044 df-recs 8105 df-rdg 8143 df-1o 8199 df-oadd 8203 df-omul 8204 df-er 8388 df-ni 10483 df-mi 10485 df-enq 10522 df-1nq 10527 |
This theorem is referenced by: recmulnq 10575 |
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