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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 7236 | . . . 4 ⊢ ~Q0 Er (ω × N) | |
2 | 1 | a1i 9 | . . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → ~Q0 Er (ω × N)) |
3 | opelxpi 4566 | . . . 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 6466 | . 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ N) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ N)) → (〈𝐴, 𝐵〉 ~Q0 〈𝐶, 𝐷〉 ↔ [〈𝐴, 𝐵〉] ~Q0 = [〈𝐶, 𝐷〉] ~Q0 )) |
6 | enq0breq 7237 | . 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 1331 ∈ wcel 1480 〈cop 3525 class class class wbr 3924 ωcom 4499 × cxp 4532 (class class class)co 5767 ·o comu 6304 Er wer 6419 [cec 6420 Ncnpi 7073 ~Q0 ceq0 7087 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-ni 7105 df-enq0 7225 |
This theorem is referenced by: nq0m0r 7257 |
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