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Mirrors > Home > MPE Home > Th. List > f1ghm0to0 | Structured version Visualization version GIF version |
Description: If a group homomorphism 𝐹 is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.) |
Ref | Expression |
---|---|
f1ghm0to0.a | ⊢ 𝐴 = (Base‘𝑅) |
f1ghm0to0.b | ⊢ 𝐵 = (Base‘𝑆) |
f1ghm0to0.n | ⊢ 𝑁 = (0g‘𝑆) |
f1ghm0to0.1 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
f1ghm0to0 | ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1ghm0to0.1 | . . . . . 6 ⊢ 0 = (0g‘𝑅) | |
2 | f1ghm0to0.n | . . . . . 6 ⊢ 𝑁 = (0g‘𝑆) | |
3 | 1, 2 | ghmid 18364 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹‘ 0 ) = 𝑁) |
4 | 3 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘ 0 ) = 𝑁) |
5 | 4 | eqeq2d 2832 | . . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘ 0 ) ↔ (𝐹‘𝑋) = 𝑁)) |
6 | simp2 1133 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴–1-1→𝐵) | |
7 | simp3 1134 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
8 | ghmgrp1 18360 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 𝑅 ∈ Grp) | |
9 | f1ghm0to0.a | . . . . . . 7 ⊢ 𝐴 = (Base‘𝑅) | |
10 | 9, 1 | grpidcl 18131 | . . . . . 6 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐴) |
11 | 8, 10 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpHom 𝑆) → 0 ∈ 𝐴) |
12 | 11 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → 0 ∈ 𝐴) |
13 | f1veqaeq 7015 | . . . 4 ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 0 ∈ 𝐴)) → ((𝐹‘𝑋) = (𝐹‘ 0 ) → 𝑋 = 0 )) | |
14 | 6, 7, 12, 13 | syl12anc 834 | . . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = (𝐹‘ 0 ) → 𝑋 = 0 )) |
15 | 5, 14 | sylbird 262 | . 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 → 𝑋 = 0 )) |
16 | fveq2 6670 | . . . 4 ⊢ (𝑋 = 0 → (𝐹‘𝑋) = (𝐹‘ 0 )) | |
17 | 16, 4 | sylan9eqr 2878 | . . 3 ⊢ (((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ∧ 𝑋 = 0 ) → (𝐹‘𝑋) = 𝑁) |
18 | 17 | ex 415 | . 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑋 = 0 → (𝐹‘𝑋) = 𝑁)) |
19 | 15, 18 | impbid 214 | 1 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 –1-1→wf1 6352 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 0gc0g 16713 Grpcgrp 18103 GrpHom cghm 18355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-ghm 18356 |
This theorem is referenced by: gim0to0 19495 kerf1ghm 19497 |
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