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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 10337 | . . 3 ⊢ ~Q Er (N × N) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ N → ~Q Er (N × N)) |
3 | mulidpi 10302 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
4 | 3, 3 | opeq12d 4804 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 = 〈𝐴, 𝐴〉) |
5 | 1pi 10299 | . . . . 5 ⊢ 1o ∈ N | |
6 | mulcanenq 10376 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N ∧ 1o ∈ N) → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) | |
7 | 5, 5, 6 | mp3an23 1449 | . . . 4 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) |
8 | df-1nq 10332 | . . . 4 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 7, 8 | breqtrrdi 5100 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 1Q) |
10 | 4, 9 | eqbrtrrd 5082 | . 2 ⊢ (𝐴 ∈ N → 〈𝐴, 𝐴〉 ~Q 1Q) |
11 | 2, 10 | ersym 8295 | 1 ⊢ (𝐴 ∈ N → 1Q ~Q 〈𝐴, 𝐴〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 × cxp 5547 (class class class)co 7150 1oc1o 8089 Er wer 8280 Ncnpi 10260 ·N cmi 10262 ~Q ceq 10267 1Qc1q 10269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ni 10288 df-mi 10290 df-enq 10327 df-1nq 10332 |
This theorem is referenced by: recmulnq 10380 |
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