odadd - Metamath Proof Explorer

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Theorem odadd 19805

Description: The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)

**Hypotheses**
Ref	Expression
odadd1.1	⊢ 𝑂 = (od‘𝐺)
odadd1.2	⊢ 𝑋 = (Base‘𝐺)
odadd1.3	⊢ + = (+_g‘𝐺)

**Assertion**
Ref	Expression
odadd	⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂‘𝐴) · (𝑂‘𝐵)))

**Proof of Theorem odadd**
Step	Hyp	Ref	Expression
1		simpl1 1189	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → 𝐺 ∈ Abel)
2		ablgrp 19740	. . . . 5 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
3	1, 2	syl 17	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → 𝐺 ∈ Grp)
4		simpl2 1190	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → 𝐴 ∈ 𝑋)
5		simpl3 1191	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → 𝐵 ∈ 𝑋)
6		odadd1.2	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
7		odadd1.3	. . . . 5 ⊢ + = (+_g‘𝐺)
8	6, 7	grpcl 18898	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
9	3, 4, 5, 8	syl3anc 1369	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝐴 + 𝐵) ∈ 𝑋)
10		odadd1.1	. . . 4 ⊢ 𝑂 = (od‘𝐺)
11	6, 10	odcl 19491	. . 3 ⊢ ((𝐴 + 𝐵) ∈ 𝑋 → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
12	9, 11	syl 17	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀)
13	6, 10	odcl 19491	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
14	4, 13	syl 17	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘𝐴) ∈ ℕ₀)
15	6, 10	odcl 19491	. . . 4 ⊢ (𝐵 ∈ 𝑋 → (𝑂‘𝐵) ∈ ℕ₀)
16	5, 15	syl 17	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘𝐵) ∈ ℕ₀)
17	14, 16	nn0mulcld 12568	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℕ₀)
18		simpr 484	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1)
19	18	oveq2d 7436	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = ((𝑂‘(𝐴 + 𝐵)) · 1))
20	12	nn0cnd 12565	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) ∈ ℂ)
21	20	mulridd 11262	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘(𝐴 + 𝐵)) · 1) = (𝑂‘(𝐴 + 𝐵)))
22	19, 21	eqtrd 2768	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) = (𝑂‘(𝐴 + 𝐵)))
23	10, 6, 7	odadd1 19803	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
24	23	adantr 480	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
25	22, 24	eqbrtrrd 5172	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
26	10, 6, 7	odadd2 19804	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))
27	26	adantr 480	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))
28	18	oveq1d 7435	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = (1↑2))
29		sq1 14191	. . . . . 6 ⊢ (1↑2) = 1
30	28, 29	eqtrdi 2784	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = 1)
31	30	oveq2d 7436	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) = ((𝑂‘(𝐴 + 𝐵)) · 1))
32	31, 21	eqtrd 2768	. . 3 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)) = (𝑂‘(𝐴 + 𝐵)))
33	27, 32	breqtrd 5174	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ (𝑂‘(𝐴 + 𝐵)))
34		dvdseq 16291	. 2 ⊢ ((((𝑂‘(𝐴 + 𝐵)) ∈ ℕ₀ ∧ ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℕ₀) ∧ ((𝑂‘(𝐴 + 𝐵)) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)) ∧ ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ (𝑂‘(𝐴 + 𝐵)))) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂‘𝐴) · (𝑂‘𝐵)))
35	12, 17, 25, 33, 34	syl22anc 838	1 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) = ((𝑂‘𝐴) · (𝑂‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ‘cfv 6548 (class class class)co 7420 1c1 11140 · cmul 11144 2c2 12298 ℕ₀cn0 12503 ↑cexp 14059 ∥ cdvds 16231 gcd cgcd 16469 Basecbs 17180 +_gcplusg 17233 Grpcgrp 18890 odcod 19479 Abelcabl 19736

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-n0 12504 df-z 12590 df-uz 12854 df-rp 13008 df-fz 13518 df-fzo 13661 df-fl 13790 df-mod 13868 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-dvds 16232 df-gcd 16470 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-od 19483 df-cmn 19737 df-abl 19738

This theorem is referenced by: gexexlem 19807

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