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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 10853 | . . 3 ⊢ ~Q Er (N × N) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝐴 ∈ N → ~Q Er (N × N)) |
3 | mulidpi 10818 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
4 | 3, 3 | opeq12d 4836 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 = 〈𝐴, 𝐴〉) |
5 | 1pi 10815 | . . . . 5 ⊢ 1o ∈ N | |
6 | mulcanenq 10892 | . . . . 5 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N ∧ 1o ∈ N) → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) | |
7 | 5, 5, 6 | mp3an23 1453 | . . . 4 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 〈1o, 1o〉) |
8 | df-1nq 10848 | . . . 4 ⊢ 1Q = 〈1o, 1o〉 | |
9 | 7, 8 | breqtrrdi 5145 | . . 3 ⊢ (𝐴 ∈ N → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 ~Q 1Q) |
10 | 4, 9 | eqbrtrrd 5127 | . 2 ⊢ (𝐴 ∈ N → 〈𝐴, 𝐴〉 ~Q 1Q) |
11 | 2, 10 | ersym 8656 | 1 ⊢ (𝐴 ∈ N → 1Q ~Q 〈𝐴, 𝐴〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 〈cop 4590 class class class wbr 5103 × cxp 5629 (class class class)co 7353 1oc1o 8401 Er wer 8641 Ncnpi 10776 ·N cmi 10778 ~Q ceq 10783 1Qc1q 10785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-oadd 8412 df-omul 8413 df-er 8644 df-ni 10804 df-mi 10806 df-enq 10843 df-1nq 10848 |
This theorem is referenced by: recmulnq 10896 |
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