Theorem conjsubgen 19180

Description: A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)

Hypotheses

Ref	Expression
conjghm.x	⊢ 𝑋 = (Base‘𝐺)
conjghm.p	⊢ + = (+g‘𝐺)
conjghm.m	⊢ − = (-g‘𝐺)
conjsubg.f	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))

Assertion

Ref	Expression
conjsubgen	⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹)

Distinct variable groups:   𝑥, −   𝑥, +   𝑥,𝐴   𝑥,𝐺   𝑥,𝑆   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem conjsubgen

Step	Hyp	Ref	Expression
1		subgrcl 19061	. . . . . . 7 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2		conjghm.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
3		conjghm.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
4		conjghm.m	. . . . . . . 8 ⊢ − = (-g‘𝐺)
5		eqid 2736	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴))
6	2, 3, 4, 5	conjghm 19178	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋))
7	1, 6	sylan 580	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋))
8		f1of1 6773	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋 → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋)
9	7, 8	simpl2im 503	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋)
10	2	subgss 19057	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
11	10	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
12		f1ssres 6737	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋 ∧ 𝑆 ⊆ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋)
13	9, 11, 12	syl2anc 584	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋)
14	11	resmptd 5999	. . . . . 6 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)))
15		conjsubg.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴))
16	14, 15	eqtr4di 2789	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹)
17		f1eq1 6725	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹 → (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋))
18	16, 17	syl 17	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋 ↔ 𝐹:𝑆–1-1→𝑋))
19	13, 18	mpbid 232	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑆–1-1→𝑋)
20		f1f1orn 6785	. . 3 ⊢ (𝐹:𝑆–1-1→𝑋 → 𝐹:𝑆–1-1-onto→ran 𝐹)
21	19, 20	syl 17	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑆–1-1-onto→ran 𝐹)
22		f1oeng 8907	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆–1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹)
23	21, 22	syldan 591	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3901   class class class wbr 5098   ↦ cmpt 5179  ran crn 5625   ↾ cres 5626  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ≈ cen 8880  Basecbs 17136  +_gcplusg 17177  Grpcgrp 18863  -_gcsg 18865  SubGrpcsubg 19050   GrpHom cghm 19141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-en 8884  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19053  df-ghm 19142

This theorem is referenced by:  slwhash  19553  sylow2  19555  sylow3lem1  19556