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Theorem mulerpqlem 9969
Description: Lemma for mulerpq 9971. (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 7365 . . . . 5 (𝐴 ∈ (N × N) → (1st𝐴) ∈ N)
213ad2ant1 1128 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐴) ∈ N)
3 xp1st 7365 . . . . 5 (𝐶 ∈ (N × N) → (1st𝐶) ∈ N)
433ad2ant3 1130 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐶) ∈ N)
5 mulclpi 9907 . . . 4 (((1st𝐴) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
62, 4, 5syl2anc 696 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (1st𝐶)) ∈ N)
7 xp2nd 7366 . . . . 5 (𝐴 ∈ (N × N) → (2nd𝐴) ∈ N)
873ad2ant1 1128 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐴) ∈ N)
9 xp2nd 7366 . . . . 5 (𝐶 ∈ (N × N) → (2nd𝐶) ∈ N)
1093ad2ant3 1130 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐶) ∈ N)
11 mulclpi 9907 . . . 4 (((2nd𝐴) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
128, 10, 11syl2anc 696 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐴) ·N (2nd𝐶)) ∈ N)
13 xp1st 7365 . . . . 5 (𝐵 ∈ (N × N) → (1st𝐵) ∈ N)
14133ad2ant2 1129 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (1st𝐵) ∈ N)
15 mulclpi 9907 . . . 4 (((1st𝐵) ∈ N ∧ (1st𝐶) ∈ N) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
1614, 4, 15syl2anc 696 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐵) ·N (1st𝐶)) ∈ N)
17 xp2nd 7366 . . . . 5 (𝐵 ∈ (N × N) → (2nd𝐵) ∈ N)
18173ad2ant2 1129 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (2nd𝐵) ∈ N)
19 mulclpi 9907 . . . 4 (((2nd𝐵) ∈ N ∧ (2nd𝐶) ∈ N) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
2018, 10, 19syl2anc 696 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((2nd𝐵) ·N (2nd𝐶)) ∈ N)
21 enqbreq 9933 . . 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 1478 . 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 9953 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
24233adant2 1126 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ·pQ 𝐶) = ⟨((1st𝐴) ·N (1st𝐶)), ((2nd𝐴) ·N (2nd𝐶))⟩)
25 mulpipq2 9953 . . . 4 ((𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
26253adant1 1125 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐵 ·pQ 𝐶) = ⟨((1st𝐵) ·N (1st𝐶)), ((2nd𝐵) ·N (2nd𝐶))⟩)
2724, 26breq12d 4817 . 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 9934 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
29283adant3 1127 . . 3 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → (𝐴 ~Q 𝐵 ↔ ((1st𝐴) ·N (2nd𝐵)) = ((1st𝐵) ·N (2nd𝐴))))
30 mulclpi 9907 . . . . 5 (((1st𝐶) ∈ N ∧ (2nd𝐶) ∈ N) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
314, 10, 30syl2anc 696 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐶) ·N (2nd𝐶)) ∈ N)
32 mulclpi 9907 . . . . 5 (((1st𝐴) ∈ N ∧ (2nd𝐵) ∈ N) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
332, 18, 32syl2anc 696 . . . 4 ((𝐴 ∈ (N × N) ∧ 𝐵 ∈ (N × N) ∧ 𝐶 ∈ (N × N)) → ((1st𝐴) ·N (2nd𝐵)) ∈ N)
34 mulcanpi 9914 . . . 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 696 . . 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 9910 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐶)))
37 fvex 6362 . . . . . . 7 (1st𝐴) ∈ V
38 fvex 6362 . . . . . . 7 (2nd𝐵) ∈ V
39 fvex 6362 . . . . . . 7 (1st𝐶) ∈ V
40 mulcompi 9910 . . . . . . 7 (𝑥 ·N 𝑦) = (𝑦 ·N 𝑥)
41 mulasspi 9911 . . . . . . 7 ((𝑥 ·N 𝑦) ·N 𝑧) = (𝑥 ·N (𝑦 ·N 𝑧))
42 fvex 6362 . . . . . . 7 (2nd𝐶) ∈ V
4337, 38, 39, 40, 41, 42caov4 7030 . . . . . 6 (((1st𝐴) ·N (2nd𝐵)) ·N ((1st𝐶) ·N (2nd𝐶))) = (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
4436, 43eqtri 2782 . . . . 5 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐴) ·N (2nd𝐵))) = (((1st𝐴) ·N (1st𝐶)) ·N ((2nd𝐵) ·N (2nd𝐶)))
45 mulcompi 9910 . . . . . 6 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶)))
46 fvex 6362 . . . . . . 7 (1st𝐵) ∈ V
47 fvex 6362 . . . . . . 7 (2nd𝐴) ∈ V
4846, 47, 39, 40, 41, 42caov4 7030 . . . . . 6 (((1st𝐵) ·N (2nd𝐴)) ·N ((1st𝐶) ·N (2nd𝐶))) = (((1st𝐵) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐶)))
49 mulcompi 9910 . . . . . 6 (((1st𝐵) ·N (1st𝐶)) ·N ((2nd𝐴) ·N (2nd𝐶))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))
5045, 48, 493eqtri 2786 . . . . 5 (((1st𝐶) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (2nd𝐴))) = (((2nd𝐴) ·N (2nd𝐶)) ·N ((1st𝐵) ·N (1st𝐶)))
5144, 50eqeq12i 2774 . . . 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 296 . 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 301 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 196  w3a 1072   = wceq 1632  wcel 2139  cop 4327   class class class wbr 4804   × cxp 5264  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  Ncnpi 9858   ·N cmi 9860   ·pQ cmpq 9863   ~Q ceq 9865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-omul 7734  df-ni 9886  df-mi 9888  df-mpq 9923  df-enq 9925
This theorem is referenced by:  mulerpq  9971
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