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Mirrors > Home > MPE Home > Th. List > enqeq | Structured version Visualization version GIF version |
Description: Corollary of nqereu 10872: if two fractions are both reduced and equivalent, then they are equal. (Contributed by Mario Carneiro, 6-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
enqeq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 1149 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → (𝐴 ∈ Q ∧ 𝐵 ∈ Q)) | |
2 | elpqn 10868 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
3 | 2 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐵 ∈ (N × N)) |
4 | nqereu 10872 | . . . 4 ⊢ (𝐵 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~Q 𝐵) | |
5 | reurmo 3359 | . . . 4 ⊢ (∃!𝑥 ∈ Q 𝑥 ~Q 𝐵 → ∃*𝑥 ∈ Q 𝑥 ~Q 𝐵) | |
6 | 3, 4, 5 | 3syl 18 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → ∃*𝑥 ∈ Q 𝑥 ~Q 𝐵) |
7 | df-rmo 3356 | . . 3 ⊢ (∃*𝑥 ∈ Q 𝑥 ~Q 𝐵 ↔ ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵)) | |
8 | 6, 7 | sylib 217 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵)) |
9 | 3simpb 1150 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → (𝐴 ∈ Q ∧ 𝐴 ~Q 𝐵)) | |
10 | simp2 1138 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐵 ∈ Q) | |
11 | enqer 10864 | . . . . 5 ⊢ ~Q Er (N × N) | |
12 | 11 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → ~Q Er (N × N)) |
13 | 12, 3 | erref 8675 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐵 ~Q 𝐵) |
14 | 10, 13 | jca 513 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → (𝐵 ∈ Q ∧ 𝐵 ~Q 𝐵)) |
15 | eleq1 2826 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Q ↔ 𝐴 ∈ Q)) | |
16 | breq1 5113 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ~Q 𝐵 ↔ 𝐴 ~Q 𝐵)) | |
17 | 15, 16 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵) ↔ (𝐴 ∈ Q ∧ 𝐴 ~Q 𝐵))) |
18 | eleq1 2826 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ Q ↔ 𝐵 ∈ Q)) | |
19 | breq1 5113 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ~Q 𝐵 ↔ 𝐵 ~Q 𝐵)) | |
20 | 18, 19 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵) ↔ (𝐵 ∈ Q ∧ 𝐵 ~Q 𝐵))) |
21 | 17, 20 | moi 3681 | . 2 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵) ∧ ((𝐴 ∈ Q ∧ 𝐴 ~Q 𝐵) ∧ (𝐵 ∈ Q ∧ 𝐵 ~Q 𝐵))) → 𝐴 = 𝐵) |
22 | 1, 8, 9, 14, 21 | syl112anc 1375 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃*wmo 2537 ∃!wreu 3354 ∃*wrmo 3355 class class class wbr 5110 × cxp 5636 Er wer 8652 Ncnpi 10787 ~Q ceq 10794 Qcnq 10795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-oadd 8421 df-omul 8422 df-er 8655 df-ni 10815 df-mi 10817 df-lti 10818 df-enq 10854 df-nq 10855 |
This theorem is referenced by: nqereq 10878 ltsonq 10912 |
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