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Theorem qusgrp2 17580

Description: Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)

**Hypotheses**
Ref	Expression
qusgrp2.u	⊢ (𝜑 → 𝑈 = (𝑅 /_s ∼ ))
qusgrp2.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusgrp2.p	⊢ (𝜑 → + = (+_g‘𝑅))
qusgrp2.r	⊢ (𝜑 → ∼ Er 𝑉)
qusgrp2.x	⊢ (𝜑 → 𝑅 ∈ 𝑋)
qusgrp2.e	⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))
qusgrp2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
qusgrp2.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) ∼ (𝑥 + (𝑦 + 𝑧)))
qusgrp2.3	⊢ (𝜑 → 0 ∈ 𝑉)
qusgrp2.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥)
qusgrp2.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉)
qusgrp2.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑁 + 𝑥) ∼ 0 )

**Assertion**
Ref	Expression
qusgrp2	⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0_g‘𝑈)))

Distinct variable groups: 𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧, ∼ 0 ,𝑎,𝑏,𝑝,𝑞,𝑥 𝑁,𝑝 𝑅,𝑝,𝑞 + ,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦 𝜑,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧 𝑉,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧 𝑈,𝑎,𝑏,𝑝,𝑞,𝑥,𝑦,𝑧 Allowed substitution hints: + (𝑧) 𝑅(𝑥,𝑦,𝑧,𝑎,𝑏) 𝑁(𝑥,𝑦,𝑧,𝑞,𝑎,𝑏) 𝑋(𝑥,𝑦,𝑧,𝑞,𝑝,𝑎,𝑏) 0 (𝑦,𝑧)

Proof of Theorem qusgrp2 Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusgrp2.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝑅 /_s ∼ ))
2		qusgrp2.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		eqid 2651	. . . 4 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusgrp2.r	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6239	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
6	2, 5	syl6eqel 2738	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 7811	. . . . 5 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V))
8	4, 6, 7	sylc 65	. . . 4 ⊢ (𝜑 → ∼ ∈ V)
9		qusgrp2.x	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
10	1, 2, 3, 8, 9	qusval 16249	. . 3 ⊢ (𝜑 → 𝑈 = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) “_s 𝑅))
11		qusgrp2.p	. . 3 ⊢ (𝜑 → + = (+_g‘𝑅))
12	1, 2, 3, 8, 9	quslem 16250	. . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ ))
13		qusgrp2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
14	13	3expb 1285	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
15		qusgrp2.e	. . . 4 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 + 𝑏) ∼ (𝑝 + 𝑞)))
16	4, 6, 3, 14, 15	ercpbl 16256	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑎) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑝) ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑏) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑞)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑎 + 𝑏)) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑝 + 𝑞))))
17	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ∼ Er 𝑉)
18		qusgrp2.2	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) ∼ (𝑥 + (𝑦 + 𝑧)))
19	17, 18	erthi 7836	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → [((𝑥 + 𝑦) + 𝑧)] ∼ = [(𝑥 + (𝑦 + 𝑧))] ∼ )
20	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 ∈ V)
21	17, 20, 3	divsfval 16254	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘((𝑥 + 𝑦) + 𝑧)) = [((𝑥 + 𝑦) + 𝑧)] ∼ )
22	17, 20, 3	divsfval 16254	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑥 + (𝑦 + 𝑧))) = [(𝑥 + (𝑦 + 𝑧))] ∼ )
23	19, 21, 22	3eqtr4d 2695	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘((𝑥 + 𝑦) + 𝑧)) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑥 + (𝑦 + 𝑧))))
24		qusgrp2.3	. . 3 ⊢ (𝜑 → 0 ∈ 𝑉)
25	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ∼ Er 𝑉)
26		qusgrp2.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥)
27	25, 26	erthi 7836	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [( 0 + 𝑥)] ∼ = [𝑥] ∼ )
28	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑉 ∈ V)
29	25, 28, 3	divsfval 16254	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘( 0 + 𝑥)) = [( 0 + 𝑥)] ∼ )
30	25, 28, 3	divsfval 16254	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑥) = [𝑥] ∼ )
31	27, 29, 30	3eqtr4d 2695	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘( 0 + 𝑥)) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑥))
32		qusgrp2.5	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑁 ∈ 𝑉)
33		qusgrp2.6	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑁 + 𝑥) ∼ 0 )
34	25, 33	ersym 7799	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∼ (𝑁 + 𝑥))
35	25, 34	erthi 7836	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [ 0 ] ∼ = [(𝑁 + 𝑥)] ∼ )
36	25, 28, 3	divsfval 16254	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ )
37	25, 28, 3	divsfval 16254	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑁 + 𝑥)) = [(𝑁 + 𝑥)] ∼ )
38	35, 36, 37	3eqtr4rd 2696	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑁 + 𝑥)) = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ))
39	10, 2, 11, 12, 16, 9, 13, 23, 24, 31, 32, 38	imasgrp2 17577	. 2 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈)))
40	4, 6, 3	divsfval 16254	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ )
41	40	eqcomd 2657	. . . 4 ⊢ (𝜑 → [ 0 ] ∼ = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ))
42	41	eqeq1d 2653	. . 3 ⊢ (𝜑 → ([ 0 ] ∼ = (0_g‘𝑈) ↔ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈)))
43	42	anbi2d 740	. 2 ⊢ (𝜑 → ((𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0_g‘𝑈)) ↔ (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈))))
44	39, 43	mpbird 247	1 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0_g‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 Vcvv 3231 class class class wbr 4685 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 Er wer 7784 [cec 7785 / cqs 7786 Basecbs 15904 +_gcplusg 15988 0_gc0g 16147 /_s cqus 16212 Grpcgrp 17469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-ec 7789 df-qs 7793 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-inf 8390 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-ip 16006 df-tset 16007 df-ple 16008 df-ds 16011 df-0g 16149 df-imas 16215 df-qus 16216 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472

This theorem is referenced by: qusgrp 17696 frgp0 18219 pi1grplem 22895

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