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Theorem nqereq 10966

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.)

**Assertion**
Ref	Expression
nqereq	⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~_Q 𝐵 ↔ ([Q]‘𝐴) = ([Q]‘𝐵)))

**Proof of Theorem nqereq**
Step	Hyp	Ref	Expression
1		nqercl 10962	. . . . 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 10962	. . . . 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 10952	. . . . . 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 10963	. . . . . . 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 8749	. . . . 5 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~_Q 𝐵) → ([Q]‘𝐴) ~_Q 𝐵)
11		nqerrel 10963	. . . . . 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 8747	. . . 4 ⊢ ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐴 ~_Q 𝐵) → ([Q]‘𝐴) ~_Q ([Q]‘𝐵))
14		enqeq 10965	. . . 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 5178	. . . 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 8750	. . 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 1533 ∈ wcel 2098 class class class wbr 5152 × cxp 5680 ‘cfv 6553 Er wer 8728 Ncnpi 10875 ~_Q ceq 10882 Qcnq 10883 [Q]cerq 10885

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-omul 8498 df-er 8731 df-ni 10903 df-mi 10905 df-lti 10906 df-enq 10942 df-nq 10943 df-erq 10944 df-1nq 10947

This theorem is referenced by: adderpq 10987 mulerpq 10988 distrnq 10992 recmulnq 10995 ltexnq 11006

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