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Theorem mndodconglem 19409

Description: Lemma for mndodcong 19410. (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)
mndodconglem.1	⊢ (𝜑 → 𝐺 ∈ Mnd)
mndodconglem.2	⊢ (𝜑 → 𝐴 ∈ 𝑋)
mndodconglem.3	⊢ (𝜑 → (𝑂‘𝐴) ∈ ℕ)
mndodconglem.4	⊢ (𝜑 → 𝑀 ∈ ℕ₀)
mndodconglem.5	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
mndodconglem.6	⊢ (𝜑 → 𝑀 < (𝑂‘𝐴))
mndodconglem.7	⊢ (𝜑 → 𝑁 < (𝑂‘𝐴))
mndodconglem.8	⊢ (𝜑 → (𝑀 · 𝐴) = (𝑁 · 𝐴))

**Assertion**
Ref	Expression
mndodconglem	⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑀 = 𝑁)

**Proof of Theorem mndodconglem**
Step	Hyp	Ref	Expression
1		mndodconglem.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
2		mndodconglem.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℕ)
3	2	nnred 12227	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℝ)
4	3	recnd 11242	. . . . . . . . 9 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℂ)
5		mndodconglem.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
6	5	nn0red 12533	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℝ)
7	6	recnd 11242	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
8		mndodconglem.5	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
9	8	nn0red 12533	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
10	9	recnd 11242	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
11	4, 7, 10	addsubassd 11591	. . . . . . . 8 ⊢ (𝜑 → (((𝑂‘𝐴) + 𝑀) − 𝑁) = ((𝑂‘𝐴) + (𝑀 − 𝑁)))
12	2	nnzd 12585	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑂‘𝐴) ∈ ℤ)
13	5	nn0zd 12584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14	12, 13	zaddcld 12670	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐴) + 𝑀) ∈ ℤ)
15	14	zred 12666	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘𝐴) + 𝑀) ∈ ℝ)
16		mndodconglem.7	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 < (𝑂‘𝐴))
17		nn0addge1 12518	. . . . . . . . . . 11 ⊢ (((𝑂‘𝐴) ∈ ℝ ∧ 𝑀 ∈ ℕ₀) → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + 𝑀))
18	3, 5, 17	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + 𝑀))
19	9, 3, 15, 16, 18	ltletrd 11374	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 < ((𝑂‘𝐴) + 𝑀))
20	8	nn0zd 12584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
21		znnsub 12608	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ ((𝑂‘𝐴) + 𝑀) ∈ ℤ) → (𝑁 < ((𝑂‘𝐴) + 𝑀) ↔ (((𝑂‘𝐴) + 𝑀) − 𝑁) ∈ ℕ))
22	20, 14, 21	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 < ((𝑂‘𝐴) + 𝑀) ↔ (((𝑂‘𝐴) + 𝑀) − 𝑁) ∈ ℕ))
23	19, 22	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (((𝑂‘𝐴) + 𝑀) − 𝑁) ∈ ℕ)
24	11, 23	eqeltrrd 2835	. . . . . . 7 ⊢ (𝜑 → ((𝑂‘𝐴) + (𝑀 − 𝑁)) ∈ ℕ)
25	4, 7, 10	addsub12d 11594	. . . . . . . . 9 ⊢ (𝜑 → ((𝑂‘𝐴) + (𝑀 − 𝑁)) = (𝑀 + ((𝑂‘𝐴) − 𝑁)))
26	25	oveq1d 7424	. . . . . . . 8 ⊢ (𝜑 → (((𝑂‘𝐴) + (𝑀 − 𝑁)) · 𝐴) = ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴))
27		mndodconglem.8	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 · 𝐴) = (𝑁 · 𝐴))
28	27	oveq1d 7424	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 · 𝐴)(+_g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴)) = ((𝑁 · 𝐴)(+_g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴)))
29		mndodconglem.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ Mnd)
30		znnsub 12608	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ (𝑂‘𝐴) ∈ ℤ) → (𝑁 < (𝑂‘𝐴) ↔ ((𝑂‘𝐴) − 𝑁) ∈ ℕ))
31	20, 12, 30	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 < (𝑂‘𝐴) ↔ ((𝑂‘𝐴) − 𝑁) ∈ ℕ))
32	16, 31	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑂‘𝐴) − 𝑁) ∈ ℕ)
33	32	nnnn0d 12532	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑂‘𝐴) − 𝑁) ∈ ℕ₀)
34		odcl.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
35		odid.3	. . . . . . . . . . . 12 ⊢ · = (._g‘𝐺)
36		eqid 2733	. . . . . . . . . . . 12 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
37	34, 35, 36	mulgnn0dir 18984	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ (𝑀 ∈ ℕ₀ ∧ ((𝑂‘𝐴) − 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋)) → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑀 · 𝐴)(+_g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴)))
38	29, 5, 33, 1, 37	syl13anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑀 · 𝐴)(+_g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴)))
39	34, 35, 36	mulgnn0dir 18984	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ Mnd ∧ (𝑁 ∈ ℕ₀ ∧ ((𝑂‘𝐴) − 𝑁) ∈ ℕ₀ ∧ 𝐴 ∈ 𝑋)) → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑁 · 𝐴)(+_g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴)))
40	29, 8, 33, 1, 39	syl13anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑁 · 𝐴)(+_g‘𝐺)(((𝑂‘𝐴) − 𝑁) · 𝐴)))
41	28, 38, 40	3eqtr4d 2783	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴))
42	10, 4	pncan3d 11574	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 + ((𝑂‘𝐴) − 𝑁)) = (𝑂‘𝐴))
43	42	oveq1d 7424	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = ((𝑂‘𝐴) · 𝐴))
44		odcl.2	. . . . . . . . . . . 12 ⊢ 𝑂 = (od‘𝐺)
45		odid.4	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝐺)
46	34, 44, 35, 45	odid 19406	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → ((𝑂‘𝐴) · 𝐴) = 0 )
47	1, 46	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑂‘𝐴) · 𝐴) = 0 )
48	43, 47	eqtrd 2773	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = 0 )
49	41, 48	eqtrd 2773	. . . . . . . 8 ⊢ (𝜑 → ((𝑀 + ((𝑂‘𝐴) − 𝑁)) · 𝐴) = 0 )
50	26, 49	eqtrd 2773	. . . . . . 7 ⊢ (𝜑 → (((𝑂‘𝐴) + (𝑀 − 𝑁)) · 𝐴) = 0 )
51	34, 44, 35, 45	odlem2 19407	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ ((𝑂‘𝐴) + (𝑀 − 𝑁)) ∈ ℕ ∧ (((𝑂‘𝐴) + (𝑀 − 𝑁)) · 𝐴) = 0 ) → (𝑂‘𝐴) ∈ (1...((𝑂‘𝐴) + (𝑀 − 𝑁))))
52	1, 24, 50, 51	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → (𝑂‘𝐴) ∈ (1...((𝑂‘𝐴) + (𝑀 − 𝑁))))
53		elfzle2 13505	. . . . . 6 ⊢ ((𝑂‘𝐴) ∈ (1...((𝑂‘𝐴) + (𝑀 − 𝑁))) → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + (𝑀 − 𝑁)))
54	52, 53	syl 17	. . . . 5 ⊢ (𝜑 → (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + (𝑀 − 𝑁)))
55	13, 20	zsubcld 12671	. . . . . . 7 ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℤ)
56	55	zred 12666	. . . . . 6 ⊢ (𝜑 → (𝑀 − 𝑁) ∈ ℝ)
57	3, 56	addge01d 11802	. . . . 5 ⊢ (𝜑 → (0 ≤ (𝑀 − 𝑁) ↔ (𝑂‘𝐴) ≤ ((𝑂‘𝐴) + (𝑀 − 𝑁))))
58	54, 57	mpbird 257	. . . 4 ⊢ (𝜑 → 0 ≤ (𝑀 − 𝑁))
59	6, 9	subge0d 11804	. . . 4 ⊢ (𝜑 → (0 ≤ (𝑀 − 𝑁) ↔ 𝑁 ≤ 𝑀))
60	58, 59	mpbid 231	. . 3 ⊢ (𝜑 → 𝑁 ≤ 𝑀)
61	6, 9	letri3d 11356	. . . 4 ⊢ (𝜑 → (𝑀 = 𝑁 ↔ (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)))
62	61	biimprd 247	. . 3 ⊢ (𝜑 → ((𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀) → 𝑀 = 𝑁))
63	60, 62	mpan2d 693	. 2 ⊢ (𝜑 → (𝑀 ≤ 𝑁 → 𝑀 = 𝑁))
64	63	imp 408	1 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑀 = 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 < clt 11248 ≤ cle 11249 − cmin 11444 ℕcn 12212 ℕ₀cn0 12472 ℤcz 12558 ...cfz 13484 Basecbs 17144 +_gcplusg 17197 0_gc0g 17385 Mndcmnd 18625 ._gcmg 18950 odcod 19392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-seq 13967 df-0g 17387 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mulg 18951 df-od 19396

This theorem is referenced by: mndodcong 19410

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