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Theorem mndodcongi 19509

Description: If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of 2 mod 10 are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)

**Hypotheses**
Ref	Expression
odcl.1	⊢ 𝑋 = (Base‘𝐺)
odcl.2	⊢ 𝑂 = (od‘𝐺)
odid.3	⊢ · = (._g‘𝐺)
odid.4	⊢ 0 = (0_g‘𝐺)

**Assertion**
Ref	Expression
mndodcongi	⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))

**Proof of Theorem mndodcongi**
Step	Hyp	Ref	Expression
1		odcl.1	. . . . . 6 ⊢ 𝑋 = (Base‘𝐺)
2		odcl.2	. . . . . 6 ⊢ 𝑂 = (od‘𝐺)
3		odid.3	. . . . . 6 ⊢ · = (._g‘𝐺)
4		odid.4	. . . . . 6 ⊢ 0 = (0_g‘𝐺)
5	1, 2, 3, 4	mndodcong 19508	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))
6	5	biimpd 230	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
7	6	3expia 1127	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
8	7	3impa 1115	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
9		nn0z 12539	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
10		nn0z 12539	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
11		zsubcl 12560	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ)
12	9, 10, 11	syl2an 602	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 − 𝑁) ∈ ℤ)
13	12	3ad2ant3 1141	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑀 − 𝑁) ∈ ℤ)
14		0dvds 16236	. . . . 5 ⊢ ((𝑀 − 𝑁) ∈ ℤ → (0 ∥ (𝑀 − 𝑁) ↔ (𝑀 − 𝑁) = 0))
15	13, 14	syl 17	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (0 ∥ (𝑀 − 𝑁) ↔ (𝑀 − 𝑁) = 0))
16		nn0cn 12438	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
17		nn0cn 12438	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
18		subeq0 11411	. . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 − 𝑁) = 0 ↔ 𝑀 = 𝑁))
19	16, 17, 18	syl2an 602	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑀 − 𝑁) = 0 ↔ 𝑀 = 𝑁))
20	19	3ad2ant3 1141	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑀 − 𝑁) = 0 ↔ 𝑀 = 𝑁))
21		oveq1 7363	. . . . 5 ⊢ (𝑀 = 𝑁 → (𝑀 · 𝐴) = (𝑁 · 𝐴))
22	20, 21	biimtrdi 254	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑀 − 𝑁) = 0 → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
23	15, 22	sylbid 241	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (0 ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
24		breq1 5075	. . . 4 ⊢ ((𝑂‘𝐴) = 0 → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) ↔ 0 ∥ (𝑀 − 𝑁)))
25	24	imbi1d 342	. . 3 ⊢ ((𝑂‘𝐴) = 0 → (((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)) ↔ (0 ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
26	23, 25	syl5ibrcom 248	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) = 0 → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
27	1, 2	odcl 19502	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
28	27	3ad2ant2 1140	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑂‘𝐴) ∈ ℕ₀)
29		elnn0 12430	. . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
30	28, 29	sylib 219	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
31	8, 26, 30	mpjaod 866	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 853 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 class class class wbr 5072 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 0cc0 11029 − cmin 11368 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ∥ cdvds 16212 Basecbs 17170 0_gc0g 17393 Mndcmnd 18693 ._gcmg 19034 odcod 19490

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fz 13453 df-fl 13742 df-mod 13820 df-seq 13955 df-dvds 16213 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mulg 19035 df-od 19494

This theorem is referenced by: (None)

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