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Mirrors > Home > ILE Home > Th. List > qdceq | GIF version |
Description: Equality of rationals is decidable. (Contributed by Jim Kingdon, 11-Oct-2021.) |
Ref | Expression |
---|---|
qdceq | ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qtri3or 10199 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴)) | |
2 | qre 9584 | . . . 4 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) | |
3 | ltne 8004 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐵 ≠ 𝐴) | |
4 | 3 | necomd 2426 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → 𝐴 ≠ 𝐵) |
5 | olc 706 | . . . . . . . 8 ⊢ (𝐴 ≠ 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
6 | dcne 2351 | . . . . . . . 8 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
7 | 5, 6 | sylibr 133 | . . . . . . 7 ⊢ (𝐴 ≠ 𝐵 → DECID 𝐴 = 𝐵) |
8 | 4, 7 | syl 14 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 𝐵) → DECID 𝐴 = 𝐵) |
9 | 8 | ex 114 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
10 | 9 | adantr 274 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
11 | 2, 10 | sylan 281 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 < 𝐵 → DECID 𝐴 = 𝐵)) |
12 | orc 707 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) | |
13 | 12, 6 | sylibr 133 | . . . 4 ⊢ (𝐴 = 𝐵 → DECID 𝐴 = 𝐵) |
14 | 13 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = 𝐵 → DECID 𝐴 = 𝐵)) |
15 | qre 9584 | . . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ) | |
16 | ltne 8004 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → 𝐴 ≠ 𝐵) | |
17 | 16, 7 | syl 14 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐵 < 𝐴) → DECID 𝐴 = 𝐵) |
18 | 17 | ex 114 | . . . . 5 ⊢ (𝐵 ∈ ℝ → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
19 | 15, 18 | syl 14 | . . . 4 ⊢ (𝐵 ∈ ℚ → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
20 | 19 | adantl 275 | . . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐵 < 𝐴 → DECID 𝐴 = 𝐵)) |
21 | 11, 14, 20 | 3jaod 1299 | . 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 < 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → DECID 𝐴 = 𝐵)) |
22 | 1, 21 | mpd 13 | 1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → DECID 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 703 DECID wdc 829 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 class class class wbr 3989 ℝcr 7773 < clt 7954 ℚcq 9578 |
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-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
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-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 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-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-n0 9136 df-z 9213 df-q 9579 df-rp 9611 |
This theorem is referenced by: flqeqceilz 10274 pcxcl 12265 pcaddlem 12292 pcadd 12293 qexpz 12304 qnnen 12386 apdifflemr 14079 |
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