Theorem conjsubgen 19282

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 19164	. . . . . . 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 2761	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) = (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴))
6	2, 3, 4, 5	conjghm 19280	. . . . . . 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 589	. . . . . 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 6800	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1-onto→𝑋 → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋)
9	7, 8	simpl2im 511	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → (𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)):𝑋–1-1→𝑋)
10	2	subgss 19160	. . . . . 6 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑋)
11	10	adantr 484	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ⊆ 𝑋)
12		f1ssres 6764	. . . . 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 593	. . . 4 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆):𝑆–1-1→𝑋)
14	11	resmptd 6025	. . . . . 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 2814	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ↦ ((𝐴 + 𝑥) − 𝐴)) ↾ 𝑆) = 𝐹)
17		f1eq1 6750	. . . . 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 234	. . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑆–1-1→𝑋)
20		f1f1orn 6813	. . 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 8945	. 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐹:𝑆–1-1-onto→ran 𝐹) → 𝑆 ≈ ran 𝐹)
23	21, 22	syldan 600	1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 ≈ ran 𝐹)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ⊆ wss 3902   class class class wbr 5097   ↦ cmpt 5178  ran crn 5644   ↾ cres 5645  –1-1→wf1 6513  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391   ≈ cen 8918  Basecbs 17236  +_gcplusg 17277  Grpcgrp 18966  -_gcsg 18968  SubGrpcsubg 19153   GrpHom cghm 19244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-map 8804  df-en 8922  df-0g 17461  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-grp 18969  df-minusg 18970  df-sbg 18971  df-subg 19156  df-ghm 19245

This theorem is referenced by:  slwhash  19655  sylow2  19657  sylow3lem1  19658