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Mirrors > Home > ILE Home > Th. List > enq0eceq | GIF version |
Description: Equivalence class equality of nonnegative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 24-Nov-2019.) |
Ref | Expression |
---|---|
enq0eceq | ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([〈𝐴, 𝐵〉] ~Q0 = [〈𝐶, 𝐷〉] ~Q0 ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enq0er 7356 | . . . 4 ⊢ ~Q0 Er (ω × N) | |
2 | 1 | a1i 9 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ~Q0 Er (ω × N)) |
3 | opelxpi 4619 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ N) → 〈𝐴, 𝐵〉 ∈ (ω × N)) | |
4 | 3 | adantr 274 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → 〈𝐴, 𝐵〉 ∈ (ω × N)) |
5 | 2, 4 | erth 6525 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q0 〈𝐶, 𝐷〉 ↔ [〈𝐴, 𝐵〉] ~Q0 = [〈𝐶, 𝐷〉] ~Q0 )) |
6 | enq0breq 7357 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q0 〈𝐶, 𝐷〉 ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶))) | |
7 | 5, 6 | bitr3d 189 | 1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ([〈𝐴, 𝐵〉] ~Q0 = [〈𝐶, 𝐷〉] ~Q0 ↔ (𝐴 ·o 𝐷) = (𝐵 ·o 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 〈cop 3563 class class class wbr 3966 ωcom 4550 × cxp 4585 (class class class)co 5825 ·o comu 6362 Er wer 6478 [cec 6479 Ncnpi 7193 ~Q0 ceq0 7207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-irdg 6318 df-oadd 6368 df-omul 6369 df-er 6481 df-ec 6483 df-ni 7225 df-enq0 7345 |
This theorem is referenced by: nq0m0r 7377 |
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