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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 2736 | . . . 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 6830 | . . . . 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 2735 | . . . 4 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
12 | 1, 3, 5, 8, 9, 10, 11 | imasgrp 19087 | . . 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 2735 | . . 3 ⊢ (Base‘(𝐹 “s 𝑊)) = (Base‘(𝐹 “s 𝑊)) | |
15 | eqid 2735 | . . 3 ⊢ (+g‘(𝐹 “s 𝑊)) = (+g‘(𝐹 “s 𝑊)) | |
16 | fof 6821 | . . . . 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 17559 | . . . . 5 ⊢ (𝜑 → (𝐹 “ 𝐵) = (Base‘(𝐹 “s 𝑊))) |
19 | 18 | feq3d 6724 | . . . 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 17579 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑥)(+g‘(𝐹 “s 𝑊))(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦))) |
22 | 21 | 3expb 1119 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑥)(+g‘(𝐹 “s 𝑊))(𝐹‘𝑦)) = (𝐹‘(𝑥 + 𝑦))) |
23 | 22 | eqcomd 2741 | . . 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 19256 | . 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 1086 = wceq 1537 ∈ wcel 2106 “ cima 5692 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 0gc0g 17486 “s cimas 17551 Grpcgrp 18964 GrpHom cghm 19243 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-imas 17555 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-ghm 19244 |
This theorem is referenced by: imaslmhm 33365 |
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