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Mirrors > Home > MPE Home > Th. List > qusgrp2 | Structured version Visualization version GIF version |
Description: Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) |
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 ) |
Ref | Expression |
---|---|
qusgrp2 | ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0g‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusgrp2.u | . . . 4 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | |
2 | qusgrp2.v | . . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | eqid 2733 | . . . 4 ⊢ (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) = (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) | |
4 | qusgrp2.r | . . . . 5 ⊢ (𝜑 → ∼ Er 𝑉) | |
5 | fvex 6856 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
6 | 2, 5 | eqeltrdi 2842 | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ V) |
7 | erex 8675 | . . . . 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 17429 | . . 3 ⊢ (𝜑 → 𝑈 = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ) “s 𝑅)) |
11 | qusgrp2.p | . . 3 ⊢ (𝜑 → + = (+g‘𝑅)) | |
12 | 1, 2, 3, 8, 9 | quslem 17430 | . . 3 ⊢ (𝜑 → (𝑢 ∈ 𝑉 ↦ [𝑢] ∼ ):𝑉–onto→(𝑉 / ∼ )) |
13 | qusgrp2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) | |
14 | 13 | 3expb 1121 | . . . 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 17436 | . . 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 482 | . . . . 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 8702 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → [((𝑥 + 𝑦) + 𝑧)] ∼ = [(𝑥 + (𝑦 + 𝑧))] ∼ ) |
20 | 6 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 ∈ V) |
21 | 17, 20, 3 | divsfval 17434 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘((𝑥 + 𝑦) + 𝑧)) = [((𝑥 + 𝑦) + 𝑧)] ∼ ) |
22 | 17, 20, 3 | divsfval 17434 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑥 + (𝑦 + 𝑧))) = [(𝑥 + (𝑦 + 𝑧))] ∼ ) |
23 | 19, 21, 22 | 3eqtr4d 2783 | . . 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 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ∼ Er 𝑉) |
26 | qusgrp2.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) ∼ 𝑥) | |
27 | 25, 26 | erthi 8702 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [( 0 + 𝑥)] ∼ = [𝑥] ∼ ) |
28 | 6 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑉 ∈ V) |
29 | 25, 28, 3 | divsfval 17434 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘( 0 + 𝑥)) = [( 0 + 𝑥)] ∼ ) |
30 | 25, 28, 3 | divsfval 17434 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘𝑥) = [𝑥] ∼ ) |
31 | 27, 29, 30 | 3eqtr4d 2783 | . . 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 8663 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ∼ (𝑁 + 𝑥)) |
35 | 25, 34 | erthi 8702 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [ 0 ] ∼ = [(𝑁 + 𝑥)] ∼ ) |
36 | 25, 28, 3 | divsfval 17434 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ ) |
37 | 25, 28, 3 | divsfval 17434 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘(𝑁 + 𝑥)) = [(𝑁 + 𝑥)] ∼ ) |
38 | 35, 36, 37 | 3eqtr4rd 2784 | . . 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 18867 | . 2 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0g‘𝑈))) |
40 | 4, 6, 3 | divsfval 17434 | . . . . 5 ⊢ (𝜑 → ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = [ 0 ] ∼ ) |
41 | 40 | eqcomd 2739 | . . . 4 ⊢ (𝜑 → [ 0 ] ∼ = ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 )) |
42 | 41 | eqeq1d 2735 | . . 3 ⊢ (𝜑 → ([ 0 ] ∼ = (0g‘𝑈) ↔ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0g‘𝑈))) |
43 | 42 | anbi2d 630 | . 2 ⊢ (𝜑 → ((𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0g‘𝑈)) ↔ (𝑈 ∈ Grp ∧ ((𝑢 ∈ 𝑉 ↦ [𝑢] ∼ )‘ 0 ) = (0g‘𝑈)))) |
44 | 39, 43 | mpbird 257 | 1 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ [ 0 ] ∼ = (0g‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3444 class class class wbr 5106 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 Er wer 8648 [cec 8649 / cqs 8650 Basecbs 17088 +gcplusg 17138 0gc0g 17326 /s cqus 17392 Grpcgrp 18753 |
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-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-ec 8653 df-qs 8657 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ds 17160 df-0g 17328 df-imas 17395 df-qus 17396 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 |
This theorem is referenced by: qusgrp 18990 frgp0 19547 pi1grplem 24428 |
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