| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > imasghm | Structured version Visualization version GIF version | ||
| Description: Given a function 𝐹 with homomorphic properties, build the image of a group. (Contributed by Thierry Arnoux, 2-Apr-2025.) |
| Ref | Expression |
|---|---|
| imasmhm.b | ⊢ 𝐵 = (Base‘𝑊) |
| imasmhm.f | ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) |
| imasmhm.1 | ⊢ + = (+g‘𝑊) |
| imasmhm.2 | ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) |
| imasghm.w | ⊢ (𝜑 → 𝑊 ∈ Grp) |
| Ref | Expression |
|---|---|
| imasghm | ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Grp ∧ 𝐹 ∈ (𝑊 GrpHom (𝐹 “s 𝑊)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqidd 2738 | . . . 4 ⊢ (𝜑 → (𝐹 “s 𝑊) = (𝐹 “s 𝑊)) | |
| 2 | imasmhm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝑊)) |
| 4 | imasmhm.1 | . . . . 5 ⊢ + = (+g‘𝑊) | |
| 5 | 4 | a1i 11 | . . . 4 ⊢ (𝜑 → + = (+g‘𝑊)) |
| 6 | imasmhm.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | |
| 7 | fimadmfo 6829 | . . . . 5 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹:𝐵–onto→(𝐹 “ 𝐵)) | |
| 8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵–onto→(𝐹 “ 𝐵)) |
| 9 | imasmhm.2 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ∧ (𝑝 ∈ 𝐵 ∧ 𝑞 ∈ 𝐵)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) | |
| 10 | imasghm.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Grp) | |
| 11 | eqid 2737 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 12 | 1, 3, 5, 8, 9, 10, 11 | imasgrp 19074 | . . 3 ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Grp ∧ (𝐹‘(0g‘𝑊)) = (0g‘(𝐹 “s 𝑊)))) |
| 13 | 12 | simpld 494 | . 2 ⊢ (𝜑 → (𝐹 “s 𝑊) ∈ Grp) |
| 14 | eqid 2737 | . . 3 ⊢ (Base‘(𝐹 “s 𝑊)) = (Base‘(𝐹 “s 𝑊)) | |
| 15 | eqid 2737 | . . 3 ⊢ (+g‘(𝐹 “s 𝑊)) = (+g‘(𝐹 “s 𝑊)) | |
| 16 | fof 6820 | . . . . 5 ⊢ (𝐹:𝐵–onto→(𝐹 “ 𝐵) → 𝐹:𝐵⟶(𝐹 “ 𝐵)) | |
| 17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(𝐹 “ 𝐵)) |
| 18 | 1, 3, 8, 10 | imasbas 17557 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐵) = (Base‘(𝐹 “s 𝑊))) |
| 19 | 18 | feq3d 6723 | . . . 4 ⊢ (𝜑 → (𝐹:𝐵⟶(𝐹 “ 𝐵) ↔ 𝐹:𝐵⟶(Base‘(𝐹 “s 𝑊)))) |
| 20 | 17, 19 | mpbid 232 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘(𝐹 “s 𝑊))) |
| 21 | 8, 9, 1, 3, 10, 4, 15 | imasaddval 17577 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑥)(+g‘(𝐹 “s 𝑊))(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦))) |
| 22 | 21 | 3expb 1121 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)(+g‘(𝐹 “s 𝑊))(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦))) |
| 23 | 22 | eqcomd 2743 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥)(+g‘(𝐹 “s 𝑊))(𝐹‘𝑦))) |
| 24 | 2, 14, 4, 15, 10, 13, 20, 23 | isghmd 19243 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝑊 GrpHom (𝐹 “s 𝑊))) |
| 25 | 13, 24 | jca 511 | 1 ⊢ (𝜑 → ((𝐹 “s 𝑊) ∈ Grp ∧ 𝐹 ∈ (𝑊 GrpHom (𝐹 “s 𝑊)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 “ cima 5688 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 “s cimas 17549 Grpcgrp 18951 GrpHom cghm 19230 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 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-nel 3047 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-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-imas 17553 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-ghm 19231 |
| This theorem is referenced by: imaslmhm 33385 |
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