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Theorem qusgrp2 18940

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 2732	. . . 4 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )
4		qusgrp2.r	. . . . 5 ⊢ (𝜑 → ∼ Er 𝑉)
5		fvex 6904	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
6	2, 5	eqeltrdi 2841	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ V)
7		erex 8726	. . . . 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 17487	. . 3 ⊢ (𝜑 → 𝑈 = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) “_s 𝑅))
11		qusgrp2.p	. . 3 ⊢ (𝜑 → + = (+_g‘𝑅))
12	1, 2, 3, 8, 9	quslem 17488	. . 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 17494	. . 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 481	. . . . 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 8753	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → [((𝑥 + 𝑦) + 𝑧)] ∼ = [(𝑥 + (𝑦 + 𝑧))] ∼ )
20	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 ∈ V)
21	17, 20, 3	divsfval 17492	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘((𝑥 + 𝑦) + 𝑧)) = [((𝑥 + 𝑦) + 𝑧)] ∼ )
22	17, 20, 3	divsfval 17492	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑥 + (𝑦 + 𝑧))) = [(𝑥 + (𝑦 + 𝑧))] ∼ )
23	19, 21, 22	3eqtr4d 2782	. . 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 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ∼ Er 𝑉)
26		qusgrp2.4	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥)
27	25, 26	erthi 8753	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [( 0 + 𝑥)] ∼ = [𝑥] ∼ )
28	6	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑉 ∈ V)
29	25, 28, 3	divsfval 17492	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘( 0 + 𝑥)) = [( 0 + 𝑥)] ∼ )
30	25, 28, 3	divsfval 17492	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑥) = [𝑥] ∼ )
31	27, 29, 30	3eqtr4d 2782	. . 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 8714	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∼ (𝑁 + 𝑥))
35	25, 34	erthi 8753	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [ 0 ] ∼ = [(𝑁 + 𝑥)] ∼ )
36	25, 28, 3	divsfval 17492	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ )
37	25, 28, 3	divsfval 17492	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑁 + 𝑥)) = [(𝑁 + 𝑥)] ∼ )
38	35, 36, 37	3eqtr4rd 2783	. . 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 18937	. 2 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈)))
40	4, 6, 3	divsfval 17492	. . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ )
41	40	eqcomd 2738	. . . 4 ⊢ (𝜑 → [ 0 ] ∼ = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ))
42	41	eqeq1d 2734	. . 3 ⊢ (𝜑 → ([ 0 ] ∼ = (0_g‘𝑈) ↔ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈)))
43	42	anbi2d 629	. 2 ⊢ (𝜑 → ((𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0_g‘𝑈)) ↔ (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0_g‘𝑈))))
44	39, 43	mpbird 256	1 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0_g‘𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7408 Er wer 8699 [cec 8700 / cqs 8701 Basecbs 17143 +_gcplusg 17196 0_gc0g 17384 /_s cqus 17450 Grpcgrp 18818

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-ec 8704 df-qs 8708 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-struct 17079 df-slot 17114 df-ndx 17126 df-base 17144 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 17386 df-imas 17453 df-qus 17454 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821

This theorem is referenced by: qusgrp 19064 frgp0 19627 pi1grplem 24564

W3C validator