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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 (𝜑𝑈 = (𝑅 /s ))
qusgrp2.v (𝜑𝑉 = (Base‘𝑅))
qusgrp2.p (𝜑+ = (+g𝑅))
qusgrp2.r (𝜑 Er 𝑉)
qusgrp2.x (𝜑𝑅𝑋)
qusgrp2.e (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 + 𝑏) (𝑝 + 𝑞)))
qusgrp2.1 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
qusgrp2.2 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) (𝑥 + (𝑦 + 𝑧)))
qusgrp2.3 (𝜑0𝑉)
qusgrp2.4 ((𝜑𝑥𝑉) → ( 0 + 𝑥) 𝑥)
qusgrp2.5 ((𝜑𝑥𝑉) → 𝑁𝑉)
qusgrp2.6 ((𝜑𝑥𝑉) → (𝑁 + 𝑥) 0 )
Assertion
Ref Expression
qusgrp2 (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] = (0g𝑈)))
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.
StepHypRef Expression
1 qusgrp2.u . . . 4 (𝜑𝑈 = (𝑅 /s ))
2 qusgrp2.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
3 eqid 2651 . . . 4 (𝑢𝑉 ↦ [𝑢] ) = (𝑢𝑉 ↦ [𝑢] )
4 qusgrp2.r . . . . 5 (𝜑 Er 𝑉)
5 fvex 6239 . . . . . 6 (Base‘𝑅) ∈ V
62, 5syl6eqel 2738 . . . . 5 (𝜑𝑉 ∈ V)
7 erex 7811 . . . . 5 ( Er 𝑉 → (𝑉 ∈ V → ∈ V))
84, 6, 7sylc 65 . . . 4 (𝜑 ∈ V)
9 qusgrp2.x . . . 4 (𝜑𝑅𝑋)
101, 2, 3, 8, 9qusval 16249 . . 3 (𝜑𝑈 = ((𝑢𝑉 ↦ [𝑢] ) “s 𝑅))
11 qusgrp2.p . . 3 (𝜑+ = (+g𝑅))
121, 2, 3, 8, 9quslem 16250 . . 3 (𝜑 → (𝑢𝑉 ↦ [𝑢] ):𝑉onto→(𝑉 / ))
13 qusgrp2.1 . . . . 5 ((𝜑𝑥𝑉𝑦𝑉) → (𝑥 + 𝑦) ∈ 𝑉)
14133expb 1285 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉)) → (𝑥 + 𝑦) ∈ 𝑉)
15 qusgrp2.e . . . 4 (𝜑 → ((𝑎 𝑝𝑏 𝑞) → (𝑎 + 𝑏) (𝑝 + 𝑞)))
164, 6, 3, 14, 15ercpbl 16256 . . 3 ((𝜑 ∧ (𝑎𝑉𝑏𝑉) ∧ (𝑝𝑉𝑞𝑉)) → ((((𝑢𝑉 ↦ [𝑢] )‘𝑎) = ((𝑢𝑉 ↦ [𝑢] )‘𝑝) ∧ ((𝑢𝑉 ↦ [𝑢] )‘𝑏) = ((𝑢𝑉 ↦ [𝑢] )‘𝑞)) → ((𝑢𝑉 ↦ [𝑢] )‘(𝑎 + 𝑏)) = ((𝑢𝑉 ↦ [𝑢] )‘(𝑝 + 𝑞))))
174adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → Er 𝑉)
18 qusgrp2.2 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑥 + 𝑦) + 𝑧) (𝑥 + (𝑦 + 𝑧)))
1917, 18erthi 7836 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → [((𝑥 + 𝑦) + 𝑧)] = [(𝑥 + (𝑦 + 𝑧))] )
206adantr 480 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → 𝑉 ∈ V)
2117, 20, 3divsfval 16254 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑢𝑉 ↦ [𝑢] )‘((𝑥 + 𝑦) + 𝑧)) = [((𝑥 + 𝑦) + 𝑧)] )
2217, 20, 3divsfval 16254 . . . 4 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑢𝑉 ↦ [𝑢] )‘(𝑥 + (𝑦 + 𝑧))) = [(𝑥 + (𝑦 + 𝑧))] )
2319, 21, 223eqtr4d 2695 . . 3 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → ((𝑢𝑉 ↦ [𝑢] )‘((𝑥 + 𝑦) + 𝑧)) = ((𝑢𝑉 ↦ [𝑢] )‘(𝑥 + (𝑦 + 𝑧))))
24 qusgrp2.3 . . 3 (𝜑0𝑉)
254adantr 480 . . . . 5 ((𝜑𝑥𝑉) → Er 𝑉)
26 qusgrp2.4 . . . . 5 ((𝜑𝑥𝑉) → ( 0 + 𝑥) 𝑥)
2725, 26erthi 7836 . . . 4 ((𝜑𝑥𝑉) → [( 0 + 𝑥)] = [𝑥] )
286adantr 480 . . . . 5 ((𝜑𝑥𝑉) → 𝑉 ∈ V)
2925, 28, 3divsfval 16254 . . . 4 ((𝜑𝑥𝑉) → ((𝑢𝑉 ↦ [𝑢] )‘( 0 + 𝑥)) = [( 0 + 𝑥)] )
3025, 28, 3divsfval 16254 . . . 4 ((𝜑𝑥𝑉) → ((𝑢𝑉 ↦ [𝑢] )‘𝑥) = [𝑥] )
3127, 29, 303eqtr4d 2695 . . 3 ((𝜑𝑥𝑉) → ((𝑢𝑉 ↦ [𝑢] )‘( 0 + 𝑥)) = ((𝑢𝑉 ↦ [𝑢] )‘𝑥))
32 qusgrp2.5 . . 3 ((𝜑𝑥𝑉) → 𝑁𝑉)
33 qusgrp2.6 . . . . . 6 ((𝜑𝑥𝑉) → (𝑁 + 𝑥) 0 )
3425, 33ersym 7799 . . . . 5 ((𝜑𝑥𝑉) → 0 (𝑁 + 𝑥))
3525, 34erthi 7836 . . . 4 ((𝜑𝑥𝑉) → [ 0 ] = [(𝑁 + 𝑥)] )
3625, 28, 3divsfval 16254 . . . 4 ((𝜑𝑥𝑉) → ((𝑢𝑉 ↦ [𝑢] )‘ 0 ) = [ 0 ] )
3725, 28, 3divsfval 16254 . . . 4 ((𝜑𝑥𝑉) → ((𝑢𝑉 ↦ [𝑢] )‘(𝑁 + 𝑥)) = [(𝑁 + 𝑥)] )
3835, 36, 373eqtr4rd 2696 . . 3 ((𝜑𝑥𝑉) → ((𝑢𝑉 ↦ [𝑢] )‘(𝑁 + 𝑥)) = ((𝑢𝑉 ↦ [𝑢] )‘ 0 ))
3910, 2, 11, 12, 16, 9, 13, 23, 24, 31, 32, 38imasgrp2 17577 . 2 (𝜑 → (𝑈 ∈ Grp ∧ ((𝑢𝑉 ↦ [𝑢] )‘ 0 ) = (0g𝑈)))
404, 6, 3divsfval 16254 . . . . 5 (𝜑 → ((𝑢𝑉 ↦ [𝑢] )‘ 0 ) = [ 0 ] )
4140eqcomd 2657 . . . 4 (𝜑 → [ 0 ] = ((𝑢𝑉 ↦ [𝑢] )‘ 0 ))
4241eqeq1d 2653 . . 3 (𝜑 → ([ 0 ] = (0g𝑈) ↔ ((𝑢𝑉 ↦ [𝑢] )‘ 0 ) = (0g𝑈)))
4342anbi2d 740 . 2 (𝜑 → ((𝑈 ∈ Grp ∧ [ 0 ] = (0g𝑈)) ↔ (𝑈 ∈ Grp ∧ ((𝑢𝑉 ↦ [𝑢] )‘ 0 ) = (0g𝑈))))
4439, 43mpbird 247 1 (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] = (0g𝑈)))
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  0gc0g 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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