modqdi - Intuitionistic Logic Explorer

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Theorem modqdi 9948

Description: Distribute multiplication over a modulo operation. (Contributed by Jim Kingdon, 26-Oct-2021.)

**Assertion**
Ref	Expression
modqdi	⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶)))

**Proof of Theorem modqdi**
Step	Hyp	Ref	Expression
1		simp1l 970	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐴 ∈ ℚ)
2		qcn 9218	. . . . 5 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
3	1, 2	syl 14	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐴 ∈ ℂ)
4		simp2 947	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐵 ∈ ℚ)
5		qcn 9218	. . . . 5 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
6	4, 5	syl 14	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐵 ∈ ℂ)
7		simp3l 974	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐶 ∈ ℚ)
8		simp3r 975	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 0 < 𝐶)
9	8	gt0ne0d 8087	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐶 ≠ 0)
10		qdivcl 9227	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ ∧ 𝐶 ≠ 0) → (𝐵 / 𝐶) ∈ ℚ)
11	4, 7, 9, 10	syl3anc 1181	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐵 / 𝐶) ∈ ℚ)
12	11	flqcld 9833	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (⌊‘(𝐵 / 𝐶)) ∈ ℤ)
13		zq 9210	. . . . . . 7 ⊢ ((⌊‘(𝐵 / 𝐶)) ∈ ℤ → (⌊‘(𝐵 / 𝐶)) ∈ ℚ)
14	12, 13	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (⌊‘(𝐵 / 𝐶)) ∈ ℚ)
15		qmulcl 9221	. . . . . 6 ⊢ ((𝐶 ∈ ℚ ∧ (⌊‘(𝐵 / 𝐶)) ∈ ℚ) → (𝐶 · (⌊‘(𝐵 / 𝐶))) ∈ ℚ)
16	7, 14, 15	syl2anc 404	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐶 · (⌊‘(𝐵 / 𝐶))) ∈ ℚ)
17		qcn 9218	. . . . 5 ⊢ ((𝐶 · (⌊‘(𝐵 / 𝐶))) ∈ ℚ → (𝐶 · (⌊‘(𝐵 / 𝐶))) ∈ ℂ)
18	16, 17	syl 14	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐶 · (⌊‘(𝐵 / 𝐶))) ∈ ℂ)
19	3, 6, 18	subdid 7989	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 − (𝐶 · (⌊‘(𝐵 / 𝐶))))) = ((𝐴 · 𝐵) − (𝐴 · (𝐶 · (⌊‘(𝐵 / 𝐶))))))
20		qcn 9218	. . . . . . . . 9 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℂ)
21	7, 20	syl 14	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐶 ∈ ℂ)
22		qre 9209	. . . . . . . . . 10 ⊢ (𝐶 ∈ ℚ → 𝐶 ∈ ℝ)
23	7, 22	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐶 ∈ ℝ)
24	23, 8	gt0ap0d 8202	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐶 # 0)
25		qre 9209	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
26	1, 25	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐴 ∈ ℝ)
27		simp1r 971	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 0 < 𝐴)
28	26, 27	gt0ap0d 8202	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 𝐴 # 0)
29	6, 21, 3, 24, 28	divcanap5d 8381	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐴 · 𝐵) / (𝐴 · 𝐶)) = (𝐵 / 𝐶))
30	29	fveq2d 5344	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (⌊‘((𝐴 · 𝐵) / (𝐴 · 𝐶))) = (⌊‘(𝐵 / 𝐶)))
31	30	oveq2d 5706	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) · (⌊‘((𝐴 · 𝐵) / (𝐴 · 𝐶)))) = ((𝐴 · 𝐶) · (⌊‘(𝐵 / 𝐶))))
32	12	zcnd 8968	. . . . . 6 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (⌊‘(𝐵 / 𝐶)) ∈ ℂ)
33	3, 21, 32	mulassd 7608	. . . . 5 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐴 · 𝐶) · (⌊‘(𝐵 / 𝐶))) = (𝐴 · (𝐶 · (⌊‘(𝐵 / 𝐶)))))
34	31, 33	eqtr2d 2128	. . . 4 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐶 · (⌊‘(𝐵 / 𝐶)))) = ((𝐴 · 𝐶) · (⌊‘((𝐴 · 𝐵) / (𝐴 · 𝐶)))))
35	34	oveq2d 5706	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐴 · 𝐵) − (𝐴 · (𝐶 · (⌊‘(𝐵 / 𝐶))))) = ((𝐴 · 𝐵) − ((𝐴 · 𝐶) · (⌊‘((𝐴 · 𝐵) / (𝐴 · 𝐶))))))
36	19, 35	eqtrd 2127	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 − (𝐶 · (⌊‘(𝐵 / 𝐶))))) = ((𝐴 · 𝐵) − ((𝐴 · 𝐶) · (⌊‘((𝐴 · 𝐵) / (𝐴 · 𝐶))))))
37		modqval 9880	. . . 4 ⊢ ((𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ ∧ 0 < 𝐶) → (𝐵 mod 𝐶) = (𝐵 − (𝐶 · (⌊‘(𝐵 / 𝐶)))))
38	4, 7, 8, 37	syl3anc 1181	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐵 mod 𝐶) = (𝐵 − (𝐶 · (⌊‘(𝐵 / 𝐶)))))
39	38	oveq2d 5706	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 mod 𝐶)) = (𝐴 · (𝐵 − (𝐶 · (⌊‘(𝐵 / 𝐶))))))
40		qmulcl 9221	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 · 𝐵) ∈ ℚ)
41	1, 4, 40	syl2anc 404	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · 𝐵) ∈ ℚ)
42		qmulcl 9221	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐶 ∈ ℚ) → (𝐴 · 𝐶) ∈ ℚ)
43	1, 7, 42	syl2anc 404	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · 𝐶) ∈ ℚ)
44	26, 23, 27, 8	mulgt0d 7703	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → 0 < (𝐴 · 𝐶))
45		modqval 9880	. . 3 ⊢ (((𝐴 · 𝐵) ∈ ℚ ∧ (𝐴 · 𝐶) ∈ ℚ ∧ 0 < (𝐴 · 𝐶)) → ((𝐴 · 𝐵) mod (𝐴 · 𝐶)) = ((𝐴 · 𝐵) − ((𝐴 · 𝐶) · (⌊‘((𝐴 · 𝐵) / (𝐴 · 𝐶))))))
46	41, 43, 44, 45	syl3anc 1181	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → ((𝐴 · 𝐵) mod (𝐴 · 𝐶)) = ((𝐴 · 𝐵) − ((𝐴 · 𝐶) · (⌊‘((𝐴 · 𝐵) / (𝐴 · 𝐶))))))
47	36, 39, 46	3eqtr4d 2137	1 ⊢ (((𝐴 ∈ ℚ ∧ 0 < 𝐴) ∧ 𝐵 ∈ ℚ ∧ (𝐶 ∈ ℚ ∧ 0 < 𝐶)) → (𝐴 · (𝐵 mod 𝐶)) = ((𝐴 · 𝐵) mod (𝐴 · 𝐶)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ≠ wne 2262 class class class wbr 3867 ‘cfv 5049 (class class class)co 5690 ℂcc 7445 ℝcr 7446 0cc0 7447 · cmul 7452 < clt 7619 − cmin 7750 / cdiv 8236 ℤcz 8848 ℚcq 9203 ⌊cfl 9824 mod cmo 9878

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 ax-arch 7561

This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-n0 8772 df-z 8849 df-q 9204 df-rp 9234 df-fl 9826 df-mod 9879

This theorem is referenced by: (None)

W3C validator