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| Mirrors > Home > MPE Home > Th. List > gim0to0 | Structured version Visualization version GIF version | ||
| Description: A group isomorphism 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, 23-May-2023.) | 
| Ref | Expression | 
|---|---|
| gim0to0.a | ⊢ 𝐴 = (Base‘𝑅) | 
| gim0to0.b | ⊢ 𝐵 = (Base‘𝑆) | 
| gim0to0.n | ⊢ 𝑁 = (0g‘𝑆) | 
| gim0to0.0 | ⊢ 0 = (0g‘𝑅) | 
| Ref | Expression | 
|---|---|
| gim0to0 | ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | gimghm 19282 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆)) | |
| 2 | gim0to0.a | . . . . . . 7 ⊢ 𝐴 = (Base‘𝑅) | |
| 3 | gim0to0.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝑆) | |
| 4 | 2, 3 | gimf1o 19281 | . . . . . 6 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴–1-1-onto→𝐵) | 
| 5 | f1of1 6847 | . . . . . 6 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) | |
| 6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝐹:𝐴–1-1→𝐵) | 
| 7 | 1, 6 | jca 511 | . . . 4 ⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵)) | 
| 8 | 7 | anim1i 615 | . . 3 ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑋 ∈ 𝐴)) | 
| 9 | df-3an 1089 | . . 3 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) ↔ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑋 ∈ 𝐴)) | |
| 10 | 8, 9 | sylibr 234 | . 2 ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴)) | 
| 11 | gim0to0.0 | . . 3 ⊢ 0 = (0g‘𝑅) | |
| 12 | gim0to0.n | . . 3 ⊢ 𝑁 = (0g‘𝑆) | |
| 13 | 2, 3, 11, 12 | f1ghm0to0 19263 | . 2 ⊢ ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | 
| 14 | 10, 13 | syl 17 | 1 ⊢ ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑁 ↔ 𝑋 = 0 )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 0gc0g 17484 GrpHom cghm 19230 GrpIso cgim 19275 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-ghm 19231 df-gim 19277 | 
| This theorem is referenced by: (None) | 
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