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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 10153 | . . . . 5 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q) | |
2 | 1 | 3ad2ant1 1113 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ∈ Q) |
3 | nqercl 10153 | . . . . 5 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q) | |
4 | 3 | 3ad2ant2 1114 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐵) ∈ Q) |
5 | enqer 10143 | . . . . . 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 10154 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴)) | |
8 | 7 | 3ad2ant1 1113 | . . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐴 ~Q ([Q]‘𝐴)) |
9 | simp3 1118 | . . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐴 ~Q 𝐵) | |
10 | 6, 8, 9 | ertr3d 8109 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ~Q 𝐵) |
11 | nqerrel 10154 | . . . . . 6 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵)) | |
12 | 11 | 3ad2ant2 1114 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐵 ~Q ([Q]‘𝐵)) |
13 | 6, 10, 12 | ertrd 8107 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ~Q ([Q]‘𝐵)) |
14 | enqeq 10156 | . . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q ∧ ([Q]‘𝐴) ~Q ([Q]‘𝐵)) → ([Q]‘𝐴) = ([Q]‘𝐵)) | |
15 | 2, 4, 13, 14 | syl3anc 1351 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) = ([Q]‘𝐵)) |
16 | 15 | 3expia 1101 | . 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 473 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q ([Q]‘𝐴)) |
19 | simprr 760 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → ([Q]‘𝐴) = ([Q]‘𝐵)) | |
20 | 18, 19 | breqtrd 4956 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q ([Q]‘𝐵)) |
21 | 11 | ad2antrl 715 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐵 ~Q ([Q]‘𝐵)) |
22 | 17, 20, 21 | ertr4d 8110 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q 𝐵) |
23 | 22 | expr 449 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐵) → 𝐴 ~Q 𝐵)) |
24 | 16, 23 | impbid 204 | 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 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 class class class wbr 4930 × cxp 5406 ‘cfv 6190 Er wer 8088 Ncnpi 10066 ~Q ceq 10073 Qcnq 10074 [Q]cerq 10076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-omul 7912 df-er 8091 df-ni 10094 df-mi 10096 df-lti 10097 df-enq 10133 df-nq 10134 df-erq 10135 df-1nq 10138 |
This theorem is referenced by: adderpq 10178 mulerpq 10179 distrnq 10183 recmulnq 10186 ltexnq 10197 |
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