mndodcongi - Metamath Proof Explorer

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Theorem mndodcongi 19561

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 19560	. . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))
6	5	biimpd 229	. . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) ∧ (𝑂‘𝐴) ∈ ℕ) → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
7	6	3expia 1122	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋) ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
8	7	3impa 1110	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∈ ℕ → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
9		nn0z 12638	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℤ)
10		nn0z 12638	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
11		zsubcl 12659	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 − 𝑁) ∈ ℤ)
12	9, 10, 11	syl2an 596	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → (𝑀 − 𝑁) ∈ ℤ)
13	12	3ad2ant3 1136	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑀 − 𝑁) ∈ ℤ)
14		0dvds 16314	. . . . 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 12536	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 𝑀 ∈ ℂ)
17		nn0cn 12536	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
18		subeq0 11535	. . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑀 − 𝑁) = 0 ↔ 𝑀 = 𝑁))
19	16, 17, 18	syl2an 596	. . . . . 6 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀) → ((𝑀 − 𝑁) = 0 ↔ 𝑀 = 𝑁))
20	19	3ad2ant3 1136	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑀 − 𝑁) = 0 ↔ 𝑀 = 𝑁))
21		oveq1 7438	. . . . 5 ⊢ (𝑀 = 𝑁 → (𝑀 · 𝐴) = (𝑁 · 𝐴))
22	20, 21	biimtrdi 253	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑀 − 𝑁) = 0 → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
23	15, 22	sylbid 240	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (0 ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
24		breq1 5146	. . . 4 ⊢ ((𝑂‘𝐴) = 0 → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) ↔ 0 ∥ (𝑀 − 𝑁)))
25	24	imbi1d 341	. . 3 ⊢ ((𝑂‘𝐴) = 0 → (((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)) ↔ (0 ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
26	23, 25	syl5ibrcom 247	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) = 0 → ((𝑂‘𝐴) ∥ (𝑀 − 𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴))))
27	1, 2	odcl 19554	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝑂‘𝐴) ∈ ℕ₀)
28	27	3ad2ant2 1135	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → (𝑂‘𝐴) ∈ ℕ₀)
29		elnn0 12528	. . 3 ⊢ ((𝑂‘𝐴) ∈ ℕ₀ ↔ ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
30	28, 29	sylib 218	. 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑋 ∧ (𝑀 ∈ ℕ₀ ∧ 𝑁 ∈ ℕ₀)) → ((𝑂‘𝐴) ∈ ℕ ∨ (𝑂‘𝐴) = 0))
31	8, 26, 30	mpjaod 861	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 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 − cmin 11492 ℕcn 12266 ℕ₀cn0 12526 ℤcz 12613 ∥ cdvds 16290 Basecbs 17247 0_gc0g 17484 Mndcmnd 18747 ._gcmg 19085 odcod 19542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 df-fl 13832 df-mod 13910 df-seq 14043 df-dvds 16291 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mulg 19086 df-od 19546

This theorem is referenced by: (None)

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