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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 10139 | . . 3 ⊢ ~Q Er (N × N) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ N → ~Q Er (N × N)) |
3 | mulidpi 10104 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
4 | 3, 3 | opeq12d 4681 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 = 〈𝐴, 𝐴〉) |
5 | 1pi 10101 | . . . . 5 ⊢ 1o ∈ N | |
6 | mulcanenq 10178 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N ∧ 1o ∈ N) → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) | |
7 | 5, 5, 6 | mp3an23 1433 | . . . 4 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) |
8 | df-1nq 10134 | . . . 4 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 7, 8 | syl6breqr 4967 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 1Q) |
10 | 4, 9 | eqbrtrrd 4949 | . 2 ⊢ (𝐴 ∈ N → 〈𝐴, 𝐴〉 ~Q 1Q) |
11 | 2, 10 | ersym 8099 | 1 ⊢ (𝐴 ∈ N → 1Q ~Q 〈𝐴, 𝐴〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 〈cop 4441 class class class wbr 4925 × cxp 5401 (class class class)co 6974 1oc1o 7896 Er wer 8084 Ncnpi 10062 ·N cmi 10064 ~Q ceq 10069 1Qc1q 10071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-omul 7908 df-er 8087 df-ni 10090 df-mi 10092 df-enq 10129 df-1nq 10134 |
This theorem is referenced by: recmulnq 10182 |
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