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Theorem ghmf1 18325
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 = (0g𝑆)
ghmf1.u 𝑈 = (0g𝑇)
Assertion
Ref Expression
ghmf1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1𝑌 ↔ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 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.
StepHypRef Expression
1 ghmf1.z . . . . . . . 8 0 = (0g𝑆)
2 ghmf1.u . . . . . . . 8 𝑈 = (0g𝑇)
31, 2ghmid 18302 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹0 ) = 𝑈)
43ad2antrr 722 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → (𝐹0 ) = 𝑈)
54eqeq2d 2829 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = (𝐹0 ) ↔ (𝐹𝑥) = 𝑈))
6 simplr 765 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → 𝐹:𝑋1-1𝑌)
7 simpr 485 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → 𝑥𝑋)
8 ghmgrp1 18298 . . . . . . . 8 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
98ad2antrr 722 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → 𝑆 ∈ Grp)
10 ghmf1.x . . . . . . . 8 𝑋 = (Base‘𝑆)
1110, 1grpidcl 18069 . . . . . . 7 (𝑆 ∈ Grp → 0𝑋)
129, 11syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → 0𝑋)
13 f1fveq 7011 . . . . . 6 ((𝐹:𝑋1-1𝑌 ∧ (𝑥𝑋0𝑋)) → ((𝐹𝑥) = (𝐹0 ) ↔ 𝑥 = 0 ))
146, 7, 12, 13syl12anc 832 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = (𝐹0 ) ↔ 𝑥 = 0 ))
155, 14bitr3d 282 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = 𝑈𝑥 = 0 ))
1615biimpd 230 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = 𝑈𝑥 = 0 ))
1716ralrimiva 3179 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) → ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 ))
18 ghmf1.y . . . . 5 𝑌 = (Base‘𝑇)
1910, 18ghmf 18300 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
2019adantr 481 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) → 𝐹:𝑋𝑌)
21 eqid 2818 . . . . . . . . . 10 (-g𝑆) = (-g𝑆)
22 eqid 2818 . . . . . . . . . 10 (-g𝑇) = (-g𝑇)
2310, 21, 22ghmsub 18304 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑦𝑋𝑧𝑋) → (𝐹‘(𝑦(-g𝑆)𝑧)) = ((𝐹𝑦)(-g𝑇)(𝐹𝑧)))
24233expb 1112 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹‘(𝑦(-g𝑆)𝑧)) = ((𝐹𝑦)(-g𝑇)(𝐹𝑧)))
2524adantlr 711 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹‘(𝑦(-g𝑆)𝑧)) = ((𝐹𝑦)(-g𝑇)(𝐹𝑧)))
2625eqeq1d 2820 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈 ↔ ((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈))
27 fveqeq2 6672 . . . . . . . 8 (𝑥 = (𝑦(-g𝑆)𝑧) → ((𝐹𝑥) = 𝑈 ↔ (𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈))
28 eqeq1 2822 . . . . . . . 8 (𝑥 = (𝑦(-g𝑆)𝑧) → (𝑥 = 0 ↔ (𝑦(-g𝑆)𝑧) = 0 ))
2927, 28imbi12d 346 . . . . . . 7 (𝑥 = (𝑦(-g𝑆)𝑧) → (((𝐹𝑥) = 𝑈𝑥 = 0 ) ↔ ((𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈 → (𝑦(-g𝑆)𝑧) = 0 )))
30 simplr 765 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 ))
318adantr 481 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) → 𝑆 ∈ Grp)
3210, 21grpsubcl 18117 . . . . . . . . 9 ((𝑆 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(-g𝑆)𝑧) ∈ 𝑋)
33323expb 1112 . . . . . . . 8 ((𝑆 ∈ Grp ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(-g𝑆)𝑧) ∈ 𝑋)
3431, 33sylan 580 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (𝑦(-g𝑆)𝑧) ∈ 𝑋)
3529, 30, 34rspcdva 3622 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈 → (𝑦(-g𝑆)𝑧) = 0 ))
3626, 35sylbird 261 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈 → (𝑦(-g𝑆)𝑧) = 0 ))
37 ghmgrp2 18299 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
3837ad2antrr 722 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑇 ∈ Grp)
3919ad2antrr 722 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝐹:𝑋𝑌)
40 simprl 767 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
4139, 40ffvelrnd 6844 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑦) ∈ 𝑌)
42 simprr 769 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
4339, 42ffvelrnd 6844 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑧) ∈ 𝑌)
4418, 2, 22grpsubeq0 18123 . . . . . 6 ((𝑇 ∈ Grp ∧ (𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑧) ∈ 𝑌) → (((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈 ↔ (𝐹𝑦) = (𝐹𝑧)))
4538, 41, 43, 44syl3anc 1363 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈 ↔ (𝐹𝑦) = (𝐹𝑧)))
468ad2antrr 722 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑆 ∈ Grp)
4710, 1, 21grpsubeq0 18123 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → ((𝑦(-g𝑆)𝑧) = 0𝑦 = 𝑧))
4846, 40, 42, 47syl3anc 1363 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(-g𝑆)𝑧) = 0𝑦 = 𝑧))
4936, 45, 483imtr3d 294 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
5049ralrimivva 3188 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) → ∀𝑦𝑋𝑧𝑋 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
51 dff13 7004 . . 3 (𝐹:𝑋1-1𝑌 ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦𝑋𝑧𝑋 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧)))
5220, 50, 51sylanbrc 583 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) → 𝐹:𝑋1-1𝑌)
5317, 52impbida 797 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1𝑌 ↔ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wf 6344  1-1wf1 6345  cfv 6348  (class class class)co 7145  Basecbs 16471  0gc0g 16701  Grpcgrp 18041  -gcsg 18043   GrpHom cghm 18293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-ghm 18294
This theorem is referenced by:  cayleylem2  18470  f1rhm0to0ALT  19423  fidomndrnglem  20007  islindf5  20911  pwssplit4  39567
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