odadd - Metamath Proof Explorer

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Theorem odadd 19881

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 1204	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → 𝐺 ∈ Abel)
2		ablgrp 19816	. . . . 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 1205	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → 𝐴 ∈ 𝑋)
5		simpl3 1206	. . . 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 18974	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 + 𝐵) ∈ 𝑋)
9	3, 4, 5, 8	syl3anc 1389	. . 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 19567	. . 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 19567	. . . 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 19567	. . . 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 12541	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∈ ℕ₀)
18		simpr 488	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1)
19	18	oveq2d 7407	. . . 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 12538	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (𝑂‘(𝐴 + 𝐵)) ∈ ℂ)
21	20	mulridd 11193	. . . 4 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘(𝐴 + 𝐵)) · 1) = (𝑂‘(𝐴 + 𝐵)))
22	19, 21	eqtrd 2796	. . 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 19879	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘(𝐴 + 𝐵)) · ((𝑂‘𝐴) gcd (𝑂‘𝐵))) ∥ ((𝑂‘𝐴) · (𝑂‘𝐵)))
24	23	adantr 484	. . 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 5121	. 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 19880	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ ((𝑂‘(𝐴 + 𝐵)) · (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2)))
27	26	adantr 484	. . 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 7406	. . . . . 6 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = (1↑2))
29		sq1 14202	. . . . . 6 ⊢ (1↑2) = 1
30	28, 29	eqtrdi 2812	. . . . 5 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → (((𝑂‘𝐴) gcd (𝑂‘𝐵))↑2) = 1)
31	30	oveq2d 7407	. . . 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 2796	. . 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 5123	. 2 ⊢ (((𝐺 ∈ Abel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ((𝑂‘𝐴) gcd (𝑂‘𝐵)) = 1) → ((𝑂‘𝐴) · (𝑂‘𝐵)) ∥ (𝑂‘(𝐴 + 𝐵)))
34		dvdseq 16339	. 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 849	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 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 class class class wbr 5097 ‘cfv 6516 (class class class)co 7391 1c1 11068 · cmul 11072 2c2 12266 ℕ₀cn0 12475 ↑cexp 14068 ∥ cdvds 16277 gcd cgcd 16519 Basecbs 17236 +_gcplusg 17277 Grpcgrp 18966 odcod 19555 Abelcabl 19812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9382 df-inf 9383 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-n0 12476 df-z 12563 df-uz 12834 df-rp 12988 df-fz 13507 df-fzo 13654 df-fl 13796 df-mod 13874 df-seq 14009 df-exp 14069 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-dvds 16278 df-gcd 16520 df-0g 17461 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18969 df-minusg 18970 df-sbg 18971 df-mulg 19101 df-od 19559 df-cmn 19813 df-abl 19814

This theorem is referenced by: gexexlem 19883

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