| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > efmndplusg | Structured version Visualization version GIF version | ||
| Description: The group operation of a monoid of endofunctions is the function composition. (Contributed by AV, 27-Jan-2024.) |
| Ref | Expression |
|---|---|
| efmndtset.g | ⊢ 𝐺 = (EndoFMnd‘𝐴) |
| efmndplusg.b | ⊢ 𝐵 = (Base‘𝐺) |
| efmndplusg.p | ⊢ + = (+g‘𝐺) |
| Ref | Expression |
|---|---|
| efmndplusg | ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | efmndtset.g | . . . . 5 ⊢ 𝐺 = (EndoFMnd‘𝐴) | |
| 2 | efmndplusg.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | 1, 2 | efmndbas 18884 | . . . . 5 ⊢ 𝐵 = (𝐴 ↑m 𝐴) |
| 4 | eqid 2737 | . . . . 5 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) | |
| 5 | eqid 2737 | . . . . 5 ⊢ (∏t‘(𝐴 × {𝒫 𝐴})) = (∏t‘(𝐴 × {𝒫 𝐴})) | |
| 6 | 1, 3, 4, 5 | efmnd 18883 | . . . 4 ⊢ (𝐴 ∈ V → 𝐺 = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉}) |
| 7 | 6 | fveq2d 6910 | . . 3 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
| 8 | efmndplusg.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 9 | 2 | fvexi 6920 | . . . . 5 ⊢ 𝐵 ∈ V |
| 10 | 9, 9 | mpoex 8104 | . . . 4 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) ∈ V |
| 11 | eqid 2737 | . . . . 5 ⊢ {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} = {〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉} | |
| 12 | 11 | topgrpplusg 17407 | . . . 4 ⊢ ((𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) ∈ V → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = (+g‘{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉})) |
| 13 | 10, 12 | ax-mp 5 | . . 3 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = (+g‘{〈(Base‘ndx), 𝐵〉, 〈(+g‘ndx), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))〉, 〈(TopSet‘ndx), (∏t‘(𝐴 × {𝒫 𝐴}))〉}) |
| 14 | 7, 8, 13 | 3eqtr4g 2802 | . 2 ⊢ (𝐴 ∈ V → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))) |
| 15 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (EndoFMnd‘𝐴) = ∅) | |
| 16 | 1, 15 | eqtrid 2789 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐺 = ∅) |
| 17 | 16 | fveq2d 6910 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (+g‘𝐺) = (+g‘∅)) |
| 18 | plusgid 17324 | . . . . 5 ⊢ +g = Slot (+g‘ndx) | |
| 19 | 18 | str0 17226 | . . . 4 ⊢ ∅ = (+g‘∅) |
| 20 | 17, 8, 19 | 3eqtr4g 2802 | . . 3 ⊢ (¬ 𝐴 ∈ V → + = ∅) |
| 21 | 16 | fveq2d 6910 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V → (Base‘𝐺) = (Base‘∅)) |
| 22 | base0 17252 | . . . . . 6 ⊢ ∅ = (Base‘∅) | |
| 23 | 21, 2, 22 | 3eqtr4g 2802 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝐵 = ∅) |
| 24 | 23 | olcd 875 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅)) |
| 25 | 0mpo0 7516 | . . . 4 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = ∅) | |
| 26 | 24, 25 | syl 17 | . . 3 ⊢ (¬ 𝐴 ∈ V → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) = ∅) |
| 27 | 20, 26 | eqtr4d 2780 | . 2 ⊢ (¬ 𝐴 ∈ V → + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))) |
| 28 | 14, 27 | pm2.61i 182 | 1 ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 𝒫 cpw 4600 {csn 4626 {ctp 4630 〈cop 4632 × cxp 5683 ∘ ccom 5689 ‘cfv 6561 ∈ cmpo 7433 ndxcnx 17230 Basecbs 17247 +gcplusg 17297 TopSetcts 17303 ∏tcpt 17483 EndoFMndcefmnd 18881 |
| 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-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-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-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-tset 17316 df-efmnd 18882 |
| This theorem is referenced by: efmndov 18894 submefmnd 18908 symgplusg 19400 efmndtmd 24109 |
| Copyright terms: Public domain | W3C validator |