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Theorem mulerpqlem 10993
Description: Lemma for mulerpq 10995. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
mulerpqlem ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶)))

Proof of Theorem mulerpqlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 8045 . . . . 5 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1132 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp1st 8045 . . . . 5 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
433ad2ant3 1134 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
5 mulclpi 10931 . . . 4 (((1st𝐴) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
62, 4, 5syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
7 xp2nd 8046 . . . . 5 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
873ad2ant1 1132 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
9 xp2nd 8046 . . . . 5 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
1093ad2ant3 1134 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
11 mulclpi 10931 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
13 xp1st 8045 . . . . 5 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
14133ad2ant2 1133 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
15 mulclpi 10931 . . . 4 (((1st𝐵) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
1614, 4, 15syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
17 xp2nd 8046 . . . . 5 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
18173ad2ant2 1133 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
19 mulclpi 10931 . . . 4 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
2018, 10, 19syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
21 enqbreq 10957 . . 3 (((((1st𝐴) ·N (1st𝐶)) ∈ N ∧ ((2nd𝐴) ·N (2nd𝐶)) ∈ N) ∧ (((1st𝐵) ·N (1st𝐶)) ∈ N ∧ ((2nd𝐵) ·N (2nd𝐶)) ∈ N)) → (⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩ ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
226, 12, 16, 20, 21syl22anc 839 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩ ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
23 mulpipq2 10977 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
24233adant2 1130 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
25 mulpipq2 10977 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
26253adant1 1129 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
2724, 26breq12d 5161 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶) ↔ ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩ ~Q ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩))
28 enqbreq2 10958 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
29283adant3 1131 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
30 mulclpi 10931 . . . . 5 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
314, 10, 30syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
32 mulclpi 10931 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
332, 18, 32syl2anc 584 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
34 mulcanpi 10938 . . . 4 ((((1st𝐶) ·N (2nd𝐶)) ∈ N ∧ ((1st𝐴) ·N (2nd𝐵)) ∈ N) → ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
3531, 33, 34syl2anc 584 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
36 mulcompi 10934 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐶)))
37 fvex 6920 . . . . . . 7 (1st𝐴) ∈ V
38 fvex 6920 . . . . . . 7 (2nd𝐵) ∈ V
39 fvex 6920 . . . . . . 7 (1st𝐶) ∈ V
40 mulcompi 10934 . . . . . . 7 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41 mulasspi 10935 . . . . . . 7 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42 fvex 6920 . . . . . . 7 (2nd𝐶) ∈ V
4337, 38, 39, 40, 41, 42caov4 7664 . . . . . 6 (((1st𝐴) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐶))) = (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
4436, 43eqtri 2763 . . . . 5 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
45 mulcompi 10934 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶)))
46 fvex 6920 . . . . . . 7 (1st𝐵) ∈ V
47 fvex 6920 . . . . . . 7 (2nd𝐴) ∈ V
4846, 47, 39, 40, 41, 42caov4 7664 . . . . . 6 (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶))) = (((1st𝐵) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐶)))
49 mulcompi 10934 . . . . . 6 (((1st𝐵) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))
5045, 48, 493eqtri 2767 . . . . 5 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))
5144, 50eqeq12i 2753 . . . 4 ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶))))
5251a1i 11 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
5329, 35, 523bitr2d 307 . 2 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))))
5422, 27, 533bitr4rd 312 1 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ (𝐴 ·pQ 𝐶) ~Q (𝐵 ·pQ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  cop 4637   class class class wbr 5148   × cxp 5687  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  Ncnpi 10882   ·N cmi 10884   ·pQ cmpq 10887   ~Q ceq 10889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-omul 8510  df-ni 10910  df-mi 10912  df-mpq 10947  df-enq 10949
This theorem is referenced by:  mulerpq  10995
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