Theorem ghmf1 18861

Description: Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
ghmf1.x	⊢ 𝑋 = (Base‘𝑆)
ghmf1.y	⊢ 𝑌 = (Base‘𝑇)
ghmf1.z	⊢ 0 = (0g‘𝑆)
ghmf1.u	⊢ 𝑈 = (0g‘𝑇)

Assertion

Ref	Expression
ghmf1	⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1→𝑌 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑇   𝑥,𝑈   𝑥,𝑋   𝑥,𝑌   𝑥, 0

Proof of Theorem ghmf1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmf1.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑆)
2		ghmf1.u	. . . . . . . 8 ⊢ 𝑈 = (0g‘𝑇)
3	1, 2	ghmid 18838	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹‘ 0 ) = 𝑈)
4	3	ad2antrr 723	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹‘ 0 ) = 𝑈)
5	4	eqeq2d 2751	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = (𝐹‘ 0 ) ↔ (𝐹‘𝑥) = 𝑈))
6		simplr 766	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋–1-1→𝑌)
7		simpr 485	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
8		ghmgrp1 18834	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
9	8	ad2antrr 723	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑆 ∈ Grp)
10		ghmf1.x	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝑆)
11	10, 1	grpidcl 18605	. . . . . . 7 ⊢ (𝑆 ∈ Grp → 0 ∈ 𝑋)
12	9, 11	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → 0 ∈ 𝑋)
13		f1fveq 7132	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (𝑥 ∈ 𝑋 ∧ 0 ∈ 𝑋)) → ((𝐹‘𝑥) = (𝐹‘ 0 ) ↔ 𝑥 = 0 ))
14	6, 7, 12, 13	syl12anc 834	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = (𝐹‘ 0 ) ↔ 𝑥 = 0 ))
15	5, 14	bitr3d 280	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 𝑈 ↔ 𝑥 = 0 ))
16	15	biimpd 228	. . 3 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ))
17	16	ralrimiva 3110	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋–1-1→𝑌) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ))
18		ghmf1.y	. . . . 5 ⊢ 𝑌 = (Base‘𝑇)
19	10, 18	ghmf 18836	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋⟶𝑌)
20	19	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → 𝐹:𝑋⟶𝑌)
21		eqid 2740	. . . . . . . . . 10 ⊢ (-g‘𝑆) = (-g‘𝑆)
22		eqid 2740	. . . . . . . . . 10 ⊢ (-g‘𝑇) = (-g‘𝑇)
23	10, 21, 22	ghmsub 18840	. . . . . . . . 9 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝐹‘(𝑦(-g‘𝑆)𝑧)) = ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)))
24	23	3expb 1119	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(-g‘𝑆)𝑧)) = ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)))
25	24	adantlr 712	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘(𝑦(-g‘𝑆)𝑧)) = ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)))
26	25	eqeq1d 2742	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈 ↔ ((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈))
27		fveqeq2 6780	. . . . . . . 8 ⊢ (𝑥 = (𝑦(-g‘𝑆)𝑧) → ((𝐹‘𝑥) = 𝑈 ↔ (𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈))
28		eqeq1 2744	. . . . . . . 8 ⊢ (𝑥 = (𝑦(-g‘𝑆)𝑧) → (𝑥 = 0 ↔ (𝑦(-g‘𝑆)𝑧) = 0 ))
29	27, 28	imbi12d 345	. . . . . . 7 ⊢ (𝑥 = (𝑦(-g‘𝑆)𝑧) → (((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ) ↔ ((𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈 → (𝑦(-g‘𝑆)𝑧) = 0 )))
30		simplr 766	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 ))
31	8	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → 𝑆 ∈ Grp)
32	10, 21	grpsubcl 18653	. . . . . . . . 9 ⊢ ((𝑆 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → (𝑦(-g‘𝑆)𝑧) ∈ 𝑋)
33	32	3expb 1119	. . . . . . . 8 ⊢ ((𝑆 ∈ Grp ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(-g‘𝑆)𝑧) ∈ 𝑋)
34	31, 33	sylan 580	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑦(-g‘𝑆)𝑧) ∈ 𝑋)
35	29, 30, 34	rspcdva 3563	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘(𝑦(-g‘𝑆)𝑧)) = 𝑈 → (𝑦(-g‘𝑆)𝑧) = 0 ))
36	26, 35	sylbird 259	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈 → (𝑦(-g‘𝑆)𝑧) = 0 ))
37		ghmgrp2 18835	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
38	37	ad2antrr 723	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑇 ∈ Grp)
39	19	ad2antrr 723	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌)
40		simprl 768	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
41	39, 40	ffvelrnd 6959	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
42		simprr 770	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑧 ∈ 𝑋)
43	39, 42	ffvelrnd 6959	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝐹‘𝑧) ∈ 𝑌)
44	18, 2, 22	grpsubeq0 18659	. . . . . 6 ⊢ ((𝑇 ∈ Grp ∧ (𝐹‘𝑦) ∈ 𝑌 ∧ (𝐹‘𝑧) ∈ 𝑌) → (((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈 ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
45	38, 41, 43, 44	syl3anc 1370	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (((𝐹‘𝑦)(-g‘𝑇)(𝐹‘𝑧)) = 𝑈 ↔ (𝐹‘𝑦) = (𝐹‘𝑧)))
46	8	ad2antrr 723	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → 𝑆 ∈ Grp)
47	10, 1, 21	grpsubeq0 18659	. . . . . 6 ⊢ ((𝑆 ∈ Grp ∧ 𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋) → ((𝑦(-g‘𝑆)𝑧) = 0 ↔ 𝑦 = 𝑧))
48	46, 40, 42, 47	syl3anc 1370	. . . . 5 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝑦(-g‘𝑆)𝑧) = 0 ↔ 𝑦 = 𝑧))
49	36, 45, 48	3imtr3d 293	. . . 4 ⊢ (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) ∧ (𝑦 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
50	49	ralrimivva 3117	. . 3 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧))
51		dff13 7125	. . 3 ⊢ (𝐹:𝑋–1-1→𝑌 ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝐹‘𝑦) = (𝐹‘𝑧) → 𝑦 = 𝑧)))
52	20, 50, 51	sylanbrc 583	. 2 ⊢ ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )) → 𝐹:𝑋–1-1→𝑌)
53	17, 52	impbida 798	1 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋–1-1→𝑌 ↔ ∀𝑥 ∈ 𝑋 ((𝐹‘𝑥) = 𝑈 → 𝑥 = 0 )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ⟶wf 6428  –1-1→wf1 6429  ‘cfv 6432  (class class class)co 7271  Basecbs 16910  0_gc0g 17148  Grpcgrp 18575  -_gcsg 18577   GrpHom cghm 18829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-0g 17150  df-mgm 18324  df-sgrp 18373  df-mnd 18384  df-grp 18578  df-minusg 18579  df-sbg 18580  df-ghm 18830

This theorem is referenced by:  cayleylem2  19019  f1rhm0to0ALT  19983  fidomndrnglem  20576  islindf5  21044  pwssplit4  40911