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Theorem grpokerinj 36756
Description: A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
grpkerinj.1 𝑋 = ran 𝐺
grpkerinj.2 𝑊 = (GId‘𝐺)
grpkerinj.3 𝑌 = ran 𝐻
grpkerinj.4 𝑈 = (GId‘𝐻)
Assertion
Ref Expression
grpokerinj ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑈}) = {𝑊}))

Proof of Theorem grpokerinj
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpkerinj.2 . . . . . . . . 9 𝑊 = (GId‘𝐺)
2 grpkerinj.4 . . . . . . . . 9 𝑈 = (GId‘𝐻)
31, 2ghomidOLD 36752 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑊) = 𝑈)
43sneqd 4640 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {(𝐹𝑊)} = {𝑈})
5 grpkerinj.1 . . . . . . . . . 10 𝑋 = ran 𝐺
6 grpkerinj.3 . . . . . . . . . 10 𝑌 = ran 𝐻
75, 6ghomf 36753 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋𝑌)
87ffnd 6718 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹 Fn 𝑋)
95, 1grpoidcl 29762 . . . . . . . . 9 (𝐺 ∈ GrpOp → 𝑊𝑋)
1093ad2ant1 1133 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑊𝑋)
11 fnsnfv 6970 . . . . . . . 8 ((𝐹 Fn 𝑋𝑊𝑋) → {(𝐹𝑊)} = (𝐹 “ {𝑊}))
128, 10, 11syl2anc 584 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {(𝐹𝑊)} = (𝐹 “ {𝑊}))
134, 12eqtr3d 2774 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑈} = (𝐹 “ {𝑊}))
1413imaeq2d 6059 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹 “ {𝑈}) = (𝐹 “ (𝐹 “ {𝑊})))
1514adantl 482 . . . 4 ((𝐹:𝑋1-1𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (𝐹 “ {𝑈}) = (𝐹 “ (𝐹 “ {𝑊})))
169snssd 4812 . . . . . 6 (𝐺 ∈ GrpOp → {𝑊} ⊆ 𝑋)
17163ad2ant1 1133 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑊} ⊆ 𝑋)
18 f1imacnv 6849 . . . . 5 ((𝐹:𝑋1-1𝑌 ∧ {𝑊} ⊆ 𝑋) → (𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
1917, 18sylan2 593 . . . 4 ((𝐹:𝑋1-1𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
2015, 19eqtrd 2772 . . 3 ((𝐹:𝑋1-1𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (𝐹 “ {𝑈}) = {𝑊})
2120expcom 414 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋1-1𝑌 → (𝐹 “ {𝑈}) = {𝑊}))
227adantr 481 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋𝑌)
23 simpl2 1192 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → 𝐻 ∈ GrpOp)
247ffvelcdmda 7086 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
2524adantrr 715 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑥) ∈ 𝑌)
267ffvelcdmda 7086 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑦𝑋) → (𝐹𝑦) ∈ 𝑌)
2726adantrl 714 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑦) ∈ 𝑌)
28 eqid 2732 . . . . . . . . 9 ( /𝑔𝐻) = ( /𝑔𝐻)
296, 2, 28grpoeqdivid 36744 . . . . . . . 8 ((𝐻 ∈ GrpOp ∧ (𝐹𝑥) ∈ 𝑌 ∧ (𝐹𝑦) ∈ 𝑌) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
3023, 25, 27, 29syl3anc 1371 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
3130adantlr 713 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
32 eqid 2732 . . . . . . . . . 10 ( /𝑔𝐺) = ( /𝑔𝐺)
335, 32, 28ghomdiv 36755 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥( /𝑔𝐺)𝑦)) = ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)))
3433adantlr 713 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥( /𝑔𝐺)𝑦)) = ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)))
3534eqeq1d 2734 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) = 𝑈 ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
362fvexi 6905 . . . . . . . . . 10 𝑈 ∈ V
3736snid 4664 . . . . . . . . 9 𝑈 ∈ {𝑈}
38 eleq1 2821 . . . . . . . . 9 ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) = 𝑈 → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ 𝑈 ∈ {𝑈}))
3937, 38mpbiri 257 . . . . . . . 8 ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) = 𝑈 → (𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈})
407ffund 6721 . . . . . . . . . . . . 13 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → Fun 𝐹)
4140adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → Fun 𝐹)
425, 32grpodivcl 29787 . . . . . . . . . . . . . . 15 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥( /𝑔𝐺)𝑦) ∈ 𝑋)
43423expb 1120 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → (𝑥( /𝑔𝐺)𝑦) ∈ 𝑋)
44433ad2antl1 1185 . . . . . . . . . . . . 13 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥( /𝑔𝐺)𝑦) ∈ 𝑋)
457fdmd 6728 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → dom 𝐹 = 𝑋)
4645adantr 481 . . . . . . . . . . . . 13 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → dom 𝐹 = 𝑋)
4744, 46eleqtrrd 2836 . . . . . . . . . . . 12 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥( /𝑔𝐺)𝑦) ∈ dom 𝐹)
48 fvimacnv 7054 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ (𝑥( /𝑔𝐺)𝑦) ∈ dom 𝐹) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ (𝐹 “ {𝑈})))
4941, 47, 48syl2anc 584 . . . . . . . . . . 11 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ (𝐹 “ {𝑈})))
50 eleq2 2822 . . . . . . . . . . 11 ((𝐹 “ {𝑈}) = {𝑊} → ((𝑥( /𝑔𝐺)𝑦) ∈ (𝐹 “ {𝑈}) ↔ (𝑥( /𝑔𝐺)𝑦) ∈ {𝑊}))
5149, 50sylan9bb 510 . . . . . . . . . 10 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ {𝑊}))
5251an32s 650 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ {𝑊}))
53 elsni 4645 . . . . . . . . . . 11 ((𝑥( /𝑔𝐺)𝑦) ∈ {𝑊} → (𝑥( /𝑔𝐺)𝑦) = 𝑊)
545, 1, 32grpoeqdivid 36744 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥 = 𝑦 ↔ (𝑥( /𝑔𝐺)𝑦) = 𝑊))
5554biimprd 247 . . . . . . . . . . . . 13 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → ((𝑥( /𝑔𝐺)𝑦) = 𝑊𝑥 = 𝑦))
56553expb 1120 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥( /𝑔𝐺)𝑦) = 𝑊𝑥 = 𝑦))
57563ad2antl1 1185 . . . . . . . . . . 11 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥( /𝑔𝐺)𝑦) = 𝑊𝑥 = 𝑦))
5853, 57syl5 34 . . . . . . . . . 10 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥( /𝑔𝐺)𝑦) ∈ {𝑊} → 𝑥 = 𝑦))
5958adantlr 713 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥( /𝑔𝐺)𝑦) ∈ {𝑊} → 𝑥 = 𝑦))
6052, 59sylbid 239 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} → 𝑥 = 𝑦))
6139, 60syl5 34 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) = 𝑈𝑥 = 𝑦))
6235, 61sylbird 259 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈𝑥 = 𝑦))
6331, 62sylbid 239 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6463ralrimivva 3200 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
65 dff13 7253 . . . 4 (𝐹:𝑋1-1𝑌 ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6622, 64, 65sylanbrc 583 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋1-1𝑌)
6766ex 413 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹 “ {𝑈}) = {𝑊} → 𝐹:𝑋1-1𝑌))
6821, 67impbid 211 1 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑈}) = {𝑊}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  wss 3948  {csn 4628  ccnv 5675  dom cdm 5676  ran crn 5677  cima 5679  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1wf1 6540  cfv 6543  (class class class)co 7408  GrpOpcgr 29737  GIdcgi 29738   /𝑔 cgs 29740   GrpOpHom cghomOLD 36746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-grpo 29741  df-gid 29742  df-ginv 29743  df-gdiv 29744  df-ghomOLD 36747
This theorem is referenced by:  rngokerinj  36838
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