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Theorem ghmf1 19037
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 19014 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹0 ) = 𝑈)
43ad2antrr 724 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → (𝐹0 ) = 𝑈)
54eqeq2d 2747 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = (𝐹0 ) ↔ (𝐹𝑥) = 𝑈))
6 simplr 767 . . . . . 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 19010 . . . . . . . 8 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
98ad2antrr 724 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → 𝑆 ∈ Grp)
10 ghmf1.x . . . . . . . 8 𝑋 = (Base‘𝑆)
1110, 1grpidcl 18778 . . . . . . 7 (𝑆 ∈ Grp → 0𝑋)
129, 11syl 17 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → 0𝑋)
13 f1fveq 7209 . . . . . 6 ((𝐹:𝑋1-1𝑌 ∧ (𝑥𝑋0𝑋)) → ((𝐹𝑥) = (𝐹0 ) ↔ 𝑥 = 0 ))
146, 7, 12, 13syl12anc 835 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = (𝐹0 ) ↔ 𝑥 = 0 ))
155, 14bitr3d 280 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = 𝑈𝑥 = 0 ))
1615biimpd 228 . . 3 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) ∧ 𝑥𝑋) → ((𝐹𝑥) = 𝑈𝑥 = 0 ))
1716ralrimiva 3143 . 2 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐹:𝑋1-1𝑌) → ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 ))
18 ghmf1.y . . . . 5 𝑌 = (Base‘𝑇)
1910, 18ghmf 19012 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:𝑋𝑌)
2019adantr 481 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) → 𝐹:𝑋𝑌)
21 eqid 2736 . . . . . . . . . 10 (-g𝑆) = (-g𝑆)
22 eqid 2736 . . . . . . . . . 10 (-g𝑇) = (-g𝑇)
2310, 21, 22ghmsub 19016 . . . . . . . . 9 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝑦𝑋𝑧𝑋) → (𝐹‘(𝑦(-g𝑆)𝑧)) = ((𝐹𝑦)(-g𝑇)(𝐹𝑧)))
24233expb 1120 . . . . . . . 8 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹‘(𝑦(-g𝑆)𝑧)) = ((𝐹𝑦)(-g𝑇)(𝐹𝑧)))
2524adantlr 713 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹‘(𝑦(-g𝑆)𝑧)) = ((𝐹𝑦)(-g𝑇)(𝐹𝑧)))
2625eqeq1d 2738 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈 ↔ ((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈))
27 fveqeq2 6851 . . . . . . . 8 (𝑥 = (𝑦(-g𝑆)𝑧) → ((𝐹𝑥) = 𝑈 ↔ (𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈))
28 eqeq1 2740 . . . . . . . 8 (𝑥 = (𝑦(-g𝑆)𝑧) → (𝑥 = 0 ↔ (𝑦(-g𝑆)𝑧) = 0 ))
2927, 28imbi12d 344 . . . . . . 7 (𝑥 = (𝑦(-g𝑆)𝑧) → (((𝐹𝑥) = 𝑈𝑥 = 0 ) ↔ ((𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈 → (𝑦(-g𝑆)𝑧) = 0 )))
30 simplr 767 . . . . . . 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 18827 . . . . . . . . 9 ((𝑆 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → (𝑦(-g𝑆)𝑧) ∈ 𝑋)
33323expb 1120 . . . . . . . 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 3582 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹‘(𝑦(-g𝑆)𝑧)) = 𝑈 → (𝑦(-g𝑆)𝑧) = 0 ))
3626, 35sylbird 259 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈 → (𝑦(-g𝑆)𝑧) = 0 ))
37 ghmgrp2 19011 . . . . . . 7 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑇 ∈ Grp)
3837ad2antrr 724 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑇 ∈ Grp)
3919ad2antrr 724 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝐹:𝑋𝑌)
40 simprl 769 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑦𝑋)
4139, 40ffvelcdmd 7036 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑦) ∈ 𝑌)
42 simprr 771 . . . . . . 7 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑧𝑋)
4339, 42ffvelcdmd 7036 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (𝐹𝑧) ∈ 𝑌)
4418, 2, 22grpsubeq0 18833 . . . . . 6 ((𝑇 ∈ Grp ∧ (𝐹𝑦) ∈ 𝑌 ∧ (𝐹𝑧) ∈ 𝑌) → (((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈 ↔ (𝐹𝑦) = (𝐹𝑧)))
4538, 41, 43, 44syl3anc 1371 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → (((𝐹𝑦)(-g𝑇)(𝐹𝑧)) = 𝑈 ↔ (𝐹𝑦) = (𝐹𝑧)))
468ad2antrr 724 . . . . . 6 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → 𝑆 ∈ Grp)
4710, 1, 21grpsubeq0 18833 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑦𝑋𝑧𝑋) → ((𝑦(-g𝑆)𝑧) = 0𝑦 = 𝑧))
4846, 40, 42, 47syl3anc 1371 . . . . 5 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝑦(-g𝑆)𝑧) = 0𝑦 = 𝑧))
4936, 45, 483imtr3d 292 . . . 4 (((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) ∧ (𝑦𝑋𝑧𝑋)) → ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
5049ralrimivva 3197 . . 3 ((𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )) → ∀𝑦𝑋𝑧𝑋 ((𝐹𝑦) = (𝐹𝑧) → 𝑦 = 𝑧))
51 dff13 7202 . . 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 799 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝐹:𝑋1-1𝑌 ↔ ∀𝑥𝑋 ((𝐹𝑥) = 𝑈𝑥 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  Basecbs 17083  0gc0g 17321  Grpcgrp 18748  -gcsg 18750   GrpHom cghm 19005
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-ghm 19006
This theorem is referenced by:  cayleylem2  19195  f1rhm0to0ALT  20175  fidomndrnglem  20777  islindf5  21245  pwssplit4  41402
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