Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > enqeq | Structured version Visualization version GIF version |
Description: Corollary of nqereu 10442: 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 10438 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 ∈ (N × N)) | |
3 | 2 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐵 ∈ (N × N)) |
4 | nqereu 10442 | . . . 4 ⊢ (𝐵 ∈ (N × N) → ∃!𝑥 ∈ Q 𝑥 ~Q 𝐵) | |
5 | reurmo 3332 | . . . 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 3062 | . . 3 ⊢ (∃*𝑥 ∈ Q 𝑥 ~Q 𝐵 ↔ ∃*𝑥(𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵)) | |
8 | 6, 7 | sylib 221 | . 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 10434 | . . . . 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 8353 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → 𝐵 ~Q 𝐵) |
14 | 10, 13 | jca 515 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q ∧ 𝐴 ~Q 𝐵) → (𝐵 ∈ Q ∧ 𝐵 ~Q 𝐵)) |
15 | eleq1 2821 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Q ↔ 𝐴 ∈ Q)) | |
16 | breq1 5043 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ~Q 𝐵 ↔ 𝐴 ~Q 𝐵)) | |
17 | 15, 16 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵) ↔ (𝐴 ∈ Q ∧ 𝐴 ~Q 𝐵))) |
18 | eleq1 2821 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ∈ Q ↔ 𝐵 ∈ Q)) | |
19 | breq1 5043 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝑥 ~Q 𝐵 ↔ 𝐵 ~Q 𝐵)) | |
20 | 18, 19 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝐵 → ((𝑥 ∈ Q ∧ 𝑥 ~Q 𝐵) ↔ (𝐵 ∈ Q ∧ 𝐵 ~Q 𝐵))) |
21 | 17, 20 | moi 3622 | . 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∃*wmo 2539 ∃!wreu 3056 ∃*wrmo 3057 class class class wbr 5040 × cxp 5533 Er wer 8330 Ncnpi 10357 ~Q ceq 10364 Qcnq 10365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 ax-un 7492 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-1st 7727 df-2nd 7728 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-oadd 8148 df-omul 8149 df-er 8333 df-ni 10385 df-mi 10387 df-lti 10388 df-enq 10424 df-nq 10425 |
This theorem is referenced by: nqereq 10448 ltsonq 10482 |
Copyright terms: Public domain | W3C validator |