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Mirrors > Home > ILE Home > Th. List > 1qec | GIF version |
Description: The equivalence class of ratio 1. (Contributed by NM, 4-Mar-1996.) |
Ref | Expression |
---|---|
1qec | ⊢ (𝐴 ∈ N → 1Q = [〈𝐴, 𝐴〉] ~Q ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-1nqqs 7300 | . 2 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
2 | 1pi 7264 | . . . 4 ⊢ 1o ∈ N | |
3 | mulcanenqec 7335 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N ∧ 1o ∈ N) → [〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉] ~Q = [〈1o, 1o〉] ~Q ) | |
4 | 2, 2, 3 | mp3an23 1324 | . . 3 ⊢ (𝐴 ∈ N → [〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉] ~Q = [〈1o, 1o〉] ~Q ) |
5 | mulidpi 7267 | . . . . 5 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
6 | 5, 5 | jca 304 | . . . 4 ⊢ (𝐴 ∈ N → ((𝐴 ·N 1o) = 𝐴 ∧ (𝐴 ·N 1o) = 𝐴)) |
7 | opeq12 3765 | . . . 4 ⊢ (((𝐴 ·N 1o) = 𝐴 ∧ (𝐴 ·N 1o) = 𝐴) → 〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 = 〈𝐴, 𝐴〉) | |
8 | eceq1 6544 | . . . 4 ⊢ (〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉 = 〈𝐴, 𝐴〉 → [〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉] ~Q = [〈𝐴, 𝐴〉] ~Q ) | |
9 | 6, 7, 8 | 3syl 17 | . . 3 ⊢ (𝐴 ∈ N → [〈(𝐴 ·N 1o), (𝐴 ·N 1o)〉] ~Q = [〈𝐴, 𝐴〉] ~Q ) |
10 | 4, 9 | eqtr3d 2205 | . 2 ⊢ (𝐴 ∈ N → [〈1o, 1o〉] ~Q = [〈𝐴, 𝐴〉] ~Q ) |
11 | 1, 10 | eqtrid 2215 | 1 ⊢ (𝐴 ∈ N → 1Q = [〈𝐴, 𝐴〉] ~Q ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 〈cop 3584 (class class class)co 5850 1oc1o 6385 [cec 6507 Ncnpi 7221 ·N cmi 7223 ~Q ceq 7228 1Qc1q 7230 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-1o 6392 df-oadd 6396 df-omul 6397 df-er 6509 df-ec 6511 df-ni 7253 df-mi 7255 df-enq 7296 df-1nqqs 7300 |
This theorem is referenced by: recexnq 7339 |
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