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Mirrors > Home > MPE Home > Th. List > nqereq | Structured version Visualization version GIF version |
Description: The function [Q] acts as a substitute for equivalence classes, and it satisfies the fundamental requirement for equivalence representatives: the representatives are equal iff the members are equivalent. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nqereq | ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ([Q]‘𝐴) = ([Q]‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqercl 10976 | . . . . 5 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q) | |
2 | 1 | 3ad2ant1 1130 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ∈ Q) |
3 | nqercl 10976 | . . . . 5 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q) | |
4 | 3 | 3ad2ant2 1131 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐵) ∈ Q) |
5 | enqer 10966 | . . . . . 6 ⊢ ~Q Er (N × N) | |
6 | 5 | a1i 11 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ~Q Er (N × N)) |
7 | nqerrel 10977 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴)) | |
8 | 7 | 3ad2ant1 1130 | . . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐴 ~Q ([Q]‘𝐴)) |
9 | simp3 1135 | . . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐴 ~Q 𝐵) | |
10 | 6, 8, 9 | ertr3d 8754 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ~Q 𝐵) |
11 | nqerrel 10977 | . . . . . 6 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵)) | |
12 | 11 | 3ad2ant2 1131 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐵 ~Q ([Q]‘𝐵)) |
13 | 6, 10, 12 | ertrd 8752 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ~Q ([Q]‘𝐵)) |
14 | enqeq 10979 | . . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q ∧ ([Q]‘𝐴) ~Q ([Q]‘𝐵)) → ([Q]‘𝐴) = ([Q]‘𝐵)) | |
15 | 2, 4, 13, 14 | syl3anc 1368 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) = ([Q]‘𝐵)) |
16 | 15 | 3expia 1118 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 → ([Q]‘𝐴) = ([Q]‘𝐵))) |
17 | 5 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → ~Q Er (N × N)) |
18 | 7 | adantr 479 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q ([Q]‘𝐴)) |
19 | simprr 771 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → ([Q]‘𝐴) = ([Q]‘𝐵)) | |
20 | 18, 19 | breqtrd 5181 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q ([Q]‘𝐵)) |
21 | 11 | ad2antrl 726 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐵 ~Q ([Q]‘𝐵)) |
22 | 17, 20, 21 | ertr4d 8755 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q 𝐵) |
23 | 22 | expr 455 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐵) → 𝐴 ~Q 𝐵)) |
24 | 16, 23 | impbid 211 | 1 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ([Q]‘𝐴) = ([Q]‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 class class class wbr 5155 × cxp 5682 ‘cfv 6556 Er wer 8733 Ncnpi 10889 ~Q ceq 10896 Qcnq 10897 [Q]cerq 10899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5306 ax-nul 5313 ax-pr 5435 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-oadd 8502 df-omul 8503 df-er 8736 df-ni 10917 df-mi 10919 df-lti 10920 df-enq 10956 df-nq 10957 df-erq 10958 df-1nq 10961 |
This theorem is referenced by: adderpq 11001 mulerpq 11002 distrnq 11006 recmulnq 11009 ltexnq 11020 |
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