qusgrp2 - Metamath Proof Explorer

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Theorem qusgrp2 18972

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 2729	. . . 4 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusgrp2.r	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6853	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
6	2, 5	eqeltrdi 2836	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 8672	. . . . 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 17481	. . 3 ⊢ (𝜑 → 𝑈 = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) “_s 𝑅))
11		qusgrp2.p	. . 3 ⊢ (𝜑 → + = (+_g‘𝑅))
12	1, 2, 3, 8, 9	quslem 17482	. . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ ))
13		qusgrp2.1	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
14	13	3expb 1120	. . . 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 17488	. . 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 8704	. . . 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 17486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘((𝑥 + 𝑦) + 𝑧)) = [((𝑥 + 𝑦) + 𝑧)] ∼ )
22	17, 20, 3	divsfval 17486	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑥 + (𝑦 + 𝑧))) = [(𝑥 + (𝑦 + 𝑧))] ∼ )
23	19, 21, 22	3eqtr4d 2774	. . 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 8704	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [( 0 + 𝑥)] ∼ = [𝑥] ∼ )
28	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑉 ∈ V)
29	25, 28, 3	divsfval 17486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘( 0 + 𝑥)) = [( 0 + 𝑥)] ∼ )
30	25, 28, 3	divsfval 17486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑥) = [𝑥] ∼ )
31	27, 29, 30	3eqtr4d 2774	. . 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 8660	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∼ (𝑁 + 𝑥))
35	25, 34	erthi 8704	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [ 0 ] ∼ = [(𝑁 + 𝑥)] ∼ )
36	25, 28, 3	divsfval 17486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ )
37	25, 28, 3	divsfval 17486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑁 + 𝑥)) = [(𝑁 + 𝑥)] ∼ )
38	35, 36, 37	3eqtr4rd 2775	. . 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 18969	. 2 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈)))
40	4, 6, 3	divsfval 17486	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ )
41	40	eqcomd 2735	. . . 4 ⊢ (𝜑 → [ 0 ] ∼ = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ))
42	41	eqeq1d 2731	. . 3 ⊢ (𝜑 → ([ 0 ] ∼ = (0_g‘𝑈) ↔ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈)))
43	42	anbi2d 630	. 2 ⊢ (𝜑 → ((𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0_g‘𝑈)) ↔ (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈))))
44	39, 43	mpbird 257	1 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0_g‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3444 class class class wbr 5102 ↦ cmpt 5183 ‘cfv 6499 (class class class)co 7369 Er wer 8645 [cec 8646 / cqs 8647 Basecbs 17155 +_gcplusg 17196 0_gc0g 17378 /_s cqus 17444 Grpcgrp 18847

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-ec 8650 df-qs 8654 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-fz 13445 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-plusg 17209 df-mulr 17210 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-0g 17380 df-imas 17447 df-qus 17448 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-grp 18850

This theorem is referenced by: qusgrp 19100 frgp0 19674 pi1grplem 24982

W3C validator