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Mirrors > Home > MPE Home > Th. List > imasgrpf1 | Structured version Visualization version GIF version |
Description: The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
imasgrpf1.u | ⊢ 𝑈 = (𝐹 “s 𝑅) |
imasgrpf1.v | ⊢ 𝑉 = (Base‘𝑅) |
Ref | Expression |
---|---|
imasgrpf1 | ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑈 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasgrpf1.u | . . . 4 ⊢ 𝑈 = (𝐹 “s 𝑅) | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑈 = (𝐹 “s 𝑅)) |
3 | imasgrpf1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑅) | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑉 = (Base‘𝑅)) |
5 | eqidd 2822 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → (+g‘𝑅) = (+g‘𝑅)) | |
6 | f1f1orn 6625 | . . . . 5 ⊢ (𝐹:𝑉–1-1→𝐵 → 𝐹:𝑉–1-1-onto→ran 𝐹) | |
7 | 6 | adantr 483 | . . . 4 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝐹:𝑉–1-1-onto→ran 𝐹) |
8 | f1ofo 6621 | . . . 4 ⊢ (𝐹:𝑉–1-1-onto→ran 𝐹 → 𝐹:𝑉–onto→ran 𝐹) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝐹:𝑉–onto→ran 𝐹) |
10 | 7 | f1ocpbl 16797 | . . 3 ⊢ (((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)))) |
11 | simpr 487 | . . 3 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑅 ∈ Grp) | |
12 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
13 | 2, 4, 5, 9, 10, 11, 12 | imasgrp 18214 | . 2 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → (𝑈 ∈ Grp ∧ (𝐹‘(0g‘𝑅)) = (0g‘𝑈))) |
14 | 13 | simpld 497 | 1 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ 𝑅 ∈ Grp) → 𝑈 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ran crn 5555 –1-1→wf1 6351 –onto→wfo 6352 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 0gc0g 16712 “s cimas 16776 Grpcgrp 18102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-fz 12892 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-plusg 16577 df-mulr 16578 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-0g 16714 df-imas 16780 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 |
This theorem is referenced by: xpsgrp 18217 |
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