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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 10971 | . . . . 5 ⊢ (𝐴 ∈ (N × N) → ([Q]‘𝐴) ∈ Q) | |
| 2 | 1 | 3ad2ant1 1134 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ∈ Q) | 
| 3 | nqercl 10971 | . . . . 5 ⊢ (𝐵 ∈ (N × N) → ([Q]‘𝐵) ∈ Q) | |
| 4 | 3 | 3ad2ant2 1135 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐵) ∈ Q) | 
| 5 | enqer 10961 | . . . . . 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 10972 | . . . . . . 7 ⊢ (𝐴 ∈ (N × N) → 𝐴 ~Q ([Q]‘𝐴)) | |
| 8 | 7 | 3ad2ant1 1134 | . . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐴 ~Q ([Q]‘𝐴)) | 
| 9 | simp3 1139 | . . . . . 6 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐴 ~Q 𝐵) | |
| 10 | 6, 8, 9 | ertr3d 8763 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ~Q 𝐵) | 
| 11 | nqerrel 10972 | . . . . . 6 ⊢ (𝐵 ∈ (N × N) → 𝐵 ~Q ([Q]‘𝐵)) | |
| 12 | 11 | 3ad2ant2 1135 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → 𝐵 ~Q ([Q]‘𝐵)) | 
| 13 | 6, 10, 12 | ertrd 8761 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) ~Q ([Q]‘𝐵)) | 
| 14 | enqeq 10974 | . . . 4 ⊢ ((([Q]‘𝐴) ∈ Q ∧ ([Q]‘𝐵) ∈ Q ∧ ([Q]‘𝐴) ~Q ([Q]‘𝐵)) → ([Q]‘𝐴) = ([Q]‘𝐵)) | |
| 15 | 2, 4, 13, 14 | syl3anc 1373 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~Q 𝐵) → ([Q]‘𝐴) = ([Q]‘𝐵)) | 
| 16 | 15 | 3expia 1122 | . 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 480 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q ([Q]‘𝐴)) | 
| 19 | simprr 773 | . . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → ([Q]‘𝐴) = ([Q]‘𝐵)) | |
| 20 | 18, 19 | breqtrd 5169 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q ([Q]‘𝐵)) | 
| 21 | 11 | ad2antrl 728 | . . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐵 ~Q ([Q]‘𝐵)) | 
| 22 | 17, 20, 21 | ertr4d 8764 | . . 3 ⊢ ((𝐴 ∈ (N × N) ∧ (𝐵 ∈ (N × N) ∧ ([Q]‘𝐴) = ([Q]‘𝐵))) → 𝐴 ~Q 𝐵) | 
| 23 | 22 | expr 456 | . 2 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (([Q]‘𝐴) = ([Q]‘𝐵) → 𝐴 ~Q 𝐵)) | 
| 24 | 16, 23 | impbid 212 | 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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 × cxp 5683 ‘cfv 6561 Er wer 8742 Ncnpi 10884 ~Q ceq 10891 Qcnq 10892 [Q]cerq 10894 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ni 10912 df-mi 10914 df-lti 10915 df-enq 10951 df-nq 10952 df-erq 10953 df-1nq 10956 | 
| This theorem is referenced by: adderpq 10996 mulerpq 10997 distrnq 11001 recmulnq 11004 ltexnq 11015 | 
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