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Theorem grpokerinj 34114
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 34110 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹𝑊) = 𝑈)
43sneqd 4346 . . . . . . 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 34111 . . . . . . . . 9 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹:𝑋𝑌)
87ffnd 6224 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝐹 Fn 𝑋)
95, 1grpoidcl 27825 . . . . . . . . 9 (𝐺 ∈ GrpOp → 𝑊𝑋)
1093ad2ant1 1163 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → 𝑊𝑋)
11 fnsnfv 6447 . . . . . . . 8 ((𝐹 Fn 𝑋𝑊𝑋) → {(𝐹𝑊)} = (𝐹 “ {𝑊}))
128, 10, 11syl2anc 579 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {(𝐹𝑊)} = (𝐹 “ {𝑊}))
134, 12eqtr3d 2801 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑈} = (𝐹 “ {𝑊}))
1413imaeq2d 5648 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹 “ {𝑈}) = (𝐹 “ (𝐹 “ {𝑊})))
1514adantl 473 . . . 4 ((𝐹:𝑋1-1𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (𝐹 “ {𝑈}) = (𝐹 “ (𝐹 “ {𝑊})))
169snssd 4494 . . . . . 6 (𝐺 ∈ GrpOp → {𝑊} ⊆ 𝑋)
17163ad2ant1 1163 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → {𝑊} ⊆ 𝑋)
18 f1imacnv 6336 . . . . 5 ((𝐹:𝑋1-1𝑌 ∧ {𝑊} ⊆ 𝑋) → (𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
1917, 18sylan2 586 . . . 4 ((𝐹:𝑋1-1𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (𝐹 “ (𝐹 “ {𝑊})) = {𝑊})
2015, 19eqtrd 2799 . . 3 ((𝐹:𝑋1-1𝑌 ∧ (𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻))) → (𝐹 “ {𝑈}) = {𝑊})
2120expcom 402 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → (𝐹:𝑋1-1𝑌 → (𝐹 “ {𝑈}) = {𝑊}))
227adantr 472 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋𝑌)
23 simpl2 1244 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → 𝐻 ∈ GrpOp)
247ffvelrnda 6549 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑥𝑋) → (𝐹𝑥) ∈ 𝑌)
2524adantrr 708 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑥) ∈ 𝑌)
267ffvelrnda 6549 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ 𝑦𝑋) → (𝐹𝑦) ∈ 𝑌)
2726adantrl 707 . . . . . . . 8 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹𝑦) ∈ 𝑌)
28 eqid 2765 . . . . . . . . 9 ( /𝑔𝐻) = ( /𝑔𝐻)
296, 2, 28grpoeqdivid 34102 . . . . . . . 8 ((𝐻 ∈ GrpOp ∧ (𝐹𝑥) ∈ 𝑌 ∧ (𝐹𝑦) ∈ 𝑌) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
3023, 25, 27, 29syl3anc 1490 . . . . . . 7 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
3130adantlr 706 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
32 eqid 2765 . . . . . . . . . 10 ( /𝑔𝐺) = ( /𝑔𝐺)
335, 32, 28ghomdiv 34113 . . . . . . . . 9 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥( /𝑔𝐺)𝑦)) = ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)))
3433adantlr 706 . . . . . . . 8 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥( /𝑔𝐺)𝑦)) = ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)))
3534eqeq1d 2767 . . . . . . 7 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) = 𝑈 ↔ ((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈))
362fvexi 6389 . . . . . . . . . 10 𝑈 ∈ V
3736snid 4366 . . . . . . . . 9 𝑈 ∈ {𝑈}
38 eleq1 2832 . . . . . . . . 9 ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) = 𝑈 → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ 𝑈 ∈ {𝑈}))
3937, 38mpbiri 249 . . . . . . . 8 ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) = 𝑈 → (𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈})
407ffund 6227 . . . . . . . . . . . . 13 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → Fun 𝐹)
4140adantr 472 . . . . . . . . . . . 12 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → Fun 𝐹)
425, 32grpodivcl 27850 . . . . . . . . . . . . . . 15 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥( /𝑔𝐺)𝑦) ∈ 𝑋)
43423expb 1149 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → (𝑥( /𝑔𝐺)𝑦) ∈ 𝑋)
44433ad2antl1 1236 . . . . . . . . . . . . 13 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥( /𝑔𝐺)𝑦) ∈ 𝑋)
457fdmd 6232 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → dom 𝐹 = 𝑋)
4645adantr 472 . . . . . . . . . . . . 13 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → dom 𝐹 = 𝑋)
4744, 46eleqtrrd 2847 . . . . . . . . . . . 12 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → (𝑥( /𝑔𝐺)𝑦) ∈ dom 𝐹)
48 fvimacnv 6522 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ (𝑥( /𝑔𝐺)𝑦) ∈ dom 𝐹) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ (𝐹 “ {𝑈})))
4941, 47, 48syl2anc 579 . . . . . . . . . . 11 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ (𝐹 “ {𝑈})))
50 eleq2 2833 . . . . . . . . . . 11 ((𝐹 “ {𝑈}) = {𝑊} → ((𝑥( /𝑔𝐺)𝑦) ∈ (𝐹 “ {𝑈}) ↔ (𝑥( /𝑔𝐺)𝑦) ∈ {𝑊}))
5149, 50sylan9bb 505 . . . . . . . . . 10 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝑥𝑋𝑦𝑋)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ {𝑊}))
5251an32s 642 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹‘(𝑥( /𝑔𝐺)𝑦)) ∈ {𝑈} ↔ (𝑥( /𝑔𝐺)𝑦) ∈ {𝑊}))
53 elsni 4351 . . . . . . . . . . 11 ((𝑥( /𝑔𝐺)𝑦) ∈ {𝑊} → (𝑥( /𝑔𝐺)𝑦) = 𝑊)
545, 1, 32grpoeqdivid 34102 . . . . . . . . . . . . . 14 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → (𝑥 = 𝑦 ↔ (𝑥( /𝑔𝐺)𝑦) = 𝑊))
5554biimprd 239 . . . . . . . . . . . . 13 ((𝐺 ∈ GrpOp ∧ 𝑥𝑋𝑦𝑋) → ((𝑥( /𝑔𝐺)𝑦) = 𝑊𝑥 = 𝑦))
56553expb 1149 . . . . . . . . . . . 12 ((𝐺 ∈ GrpOp ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥( /𝑔𝐺)𝑦) = 𝑊𝑥 = 𝑦))
57563ad2antl1 1236 . . . . . . . . . . 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 706 . . . . . . . . 9 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝑥( /𝑔𝐺)𝑦) ∈ {𝑊} → 𝑥 = 𝑦))
6052, 59sylbid 231 . . . . . . . 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 251 . . . . . 6 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → (((𝐹𝑥)( /𝑔𝐻)(𝐹𝑦)) = 𝑈𝑥 = 𝑦))
6331, 62sylbid 231 . . . . 5 ((((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
6463ralrimivva 3118 . . . 4 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦))
65 dff13 6704 . . . 4 (𝐹:𝑋1-1𝑌 ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
6622, 64, 65sylanbrc 578 . . 3 (((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) ∧ (𝐹 “ {𝑈}) = {𝑊}) → 𝐹:𝑋1-1𝑌)
6766ex 401 . 2 ((𝐺 ∈ GrpOp ∧ 𝐻 ∈ GrpOp ∧ 𝐹 ∈ (𝐺 GrpOpHom 𝐻)) → ((𝐹 “ {𝑈}) = {𝑊} → 𝐹:𝑋1-1𝑌))
6821, 67impbid 203 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wss 3732  {csn 4334  ccnv 5276  dom cdm 5277  ran crn 5278  cima 5280  Fun wfun 6062   Fn wfn 6063  wf 6064  1-1wf1 6065  cfv 6068  (class class class)co 6842  GrpOpcgr 27800  GIdcgi 27801   /𝑔 cgs 27803   GrpOpHom cghomOLD 34104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-grpo 27804  df-gid 27805  df-ginv 27806  df-gdiv 27807  df-ghomOLD 34105
This theorem is referenced by:  rngokerinj  34196
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