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| Mirrors > Home > MPE Home > Th. List > ghmquskerco | Structured version Visualization version GIF version | ||
| Description: In the case of theorem ghmqusker 19305, the composition of the natural homomorphism 𝐿 with the constructed homomorphism 𝐽 equals the original homomorphism 𝐹. One says that 𝐹 factors through 𝑄. (Proposed by Saveliy Skresanov, 15-Feb-2025.) (Contributed by Thierry Arnoux, 15-Feb-2025.) |
| Ref | Expression |
|---|---|
| ghmqusker.1 | ⊢ 0 = (0g‘𝐻) |
| ghmqusker.f | ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
| ghmqusker.k | ⊢ 𝐾 = (◡𝐹 “ { 0 }) |
| ghmqusker.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) |
| ghmqusker.j | ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) |
| ghmquskerco.b | ⊢ 𝐵 = (Base‘𝐺) |
| ghmquskerco.l | ⊢ 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝐾)) |
| Ref | Expression |
|---|---|
| ghmquskerco | ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ghmqusker.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻)) | |
| 2 | ghmquskerco.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 4 | 2, 3 | ghmf 19238 | . . . 4 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐹:𝐵⟶(Base‘𝐻)) |
| 5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐻)) |
| 6 | 5 | ffnd 6737 | . 2 ⊢ (𝜑 → 𝐹 Fn 𝐵) |
| 7 | 1 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹 ∈ (𝐺 GrpHom 𝐻)) |
| 8 | 7 | imaexd 7938 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹 “ [𝑥](𝐺 ~QG 𝐾)) ∈ V) |
| 9 | 8 | uniexd 7762 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾)) ∈ V) |
| 10 | 9 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾)) ∈ V) |
| 11 | eqid 2737 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾))) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾))) | |
| 12 | 11 | fnmpt 6708 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾)) ∈ V → (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾))) Fn 𝐵) |
| 13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾))) Fn 𝐵) |
| 14 | ovex 7464 | . . . . . . . 8 ⊢ (𝐺 ~QG 𝐾) ∈ V | |
| 15 | 14 | ecelqsi 8813 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐵 → [𝑥](𝐺 ~QG 𝐾) ∈ (𝐵 / (𝐺 ~QG 𝐾))) |
| 16 | 15 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥](𝐺 ~QG 𝐾) ∈ (𝐵 / (𝐺 ~QG 𝐾))) |
| 17 | ghmqusker.q | . . . . . . . . 9 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)) | |
| 18 | 17 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))) |
| 19 | 2 | a1i 11 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
| 20 | ovexd 7466 | . . . . . . . 8 ⊢ (𝜑 → (𝐺 ~QG 𝐾) ∈ V) | |
| 21 | reldmghm 19232 | . . . . . . . . . . 11 ⊢ Rel dom GrpHom | |
| 22 | 21 | ovrcl 7472 | . . . . . . . . . 10 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V)) |
| 23 | 22 | simpld 494 | . . . . . . . . 9 ⊢ (𝐹 ∈ (𝐺 GrpHom 𝐻) → 𝐺 ∈ V) |
| 24 | 1, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ V) |
| 25 | 18, 19, 20, 24 | qusbas 17590 | . . . . . . 7 ⊢ (𝜑 → (𝐵 / (𝐺 ~QG 𝐾)) = (Base‘𝑄)) |
| 26 | 25 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐵 / (𝐺 ~QG 𝐾)) = (Base‘𝑄)) |
| 27 | 16, 26 | eleqtrd 2843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥](𝐺 ~QG 𝐾) ∈ (Base‘𝑄)) |
| 28 | ghmquskerco.l | . . . . . 6 ⊢ 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝐾)) | |
| 29 | 28 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝐾))) |
| 30 | ghmqusker.j | . . . . . 6 ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞)) | |
| 31 | 30 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))) |
| 32 | imaeq2 6074 | . . . . . 6 ⊢ (𝑞 = [𝑥](𝐺 ~QG 𝐾) → (𝐹 “ 𝑞) = (𝐹 “ [𝑥](𝐺 ~QG 𝐾))) | |
| 33 | 32 | unieqd 4920 | . . . . 5 ⊢ (𝑞 = [𝑥](𝐺 ~QG 𝐾) → ∪ (𝐹 “ 𝑞) = ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾))) |
| 34 | 27, 29, 31, 33 | fmptco 7149 | . . . 4 ⊢ (𝜑 → (𝐽 ∘ 𝐿) = (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾)))) |
| 35 | 34 | fneq1d 6661 | . . 3 ⊢ (𝜑 → ((𝐽 ∘ 𝐿) Fn 𝐵 ↔ (𝑥 ∈ 𝐵 ↦ ∪ (𝐹 “ [𝑥](𝐺 ~QG 𝐾))) Fn 𝐵)) |
| 36 | 13, 35 | mpbird 257 | . 2 ⊢ (𝜑 → (𝐽 ∘ 𝐿) Fn 𝐵) |
| 37 | ecexg 8749 | . . . . . 6 ⊢ ((𝐺 ~QG 𝐾) ∈ V → [𝑥](𝐺 ~QG 𝐾) ∈ V) | |
| 38 | 14, 37 | ax-mp 5 | . . . . 5 ⊢ [𝑥](𝐺 ~QG 𝐾) ∈ V |
| 39 | 38, 28 | fnmpti 6711 | . . . 4 ⊢ 𝐿 Fn 𝐵 |
| 40 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
| 41 | fvco2 7006 | . . . 4 ⊢ ((𝐿 Fn 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝐽 ∘ 𝐿)‘𝑥) = (𝐽‘(𝐿‘𝑥))) | |
| 42 | 39, 40, 41 | sylancr 587 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐽 ∘ 𝐿)‘𝑥) = (𝐽‘(𝐿‘𝑥))) |
| 43 | 38 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → [𝑥](𝐺 ~QG 𝐾) ∈ V) |
| 44 | 29, 43 | fvmpt2d 7029 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐿‘𝑥) = [𝑥](𝐺 ~QG 𝐾)) |
| 45 | 44 | fveq2d 6910 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘(𝐿‘𝑥)) = (𝐽‘[𝑥](𝐺 ~QG 𝐾))) |
| 46 | ghmqusker.1 | . . . 4 ⊢ 0 = (0g‘𝐻) | |
| 47 | ghmqusker.k | . . . 4 ⊢ 𝐾 = (◡𝐹 “ { 0 }) | |
| 48 | 40, 2 | eleqtrdi 2851 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ (Base‘𝐺)) |
| 49 | 46, 7, 47, 17, 30, 48 | ghmquskerlem1 19301 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐽‘[𝑥](𝐺 ~QG 𝐾)) = (𝐹‘𝑥)) |
| 50 | 42, 45, 49 | 3eqtrrd 2782 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = ((𝐽 ∘ 𝐿)‘𝑥)) |
| 51 | 6, 36, 50 | eqfnfvd 7054 | 1 ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 ◡ccnv 5684 “ cima 5688 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 [cec 8743 / cqs 8744 Basecbs 17247 0gc0g 17484 /s cqus 17550 ~QG cqg 19140 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-ec 8747 df-qs 8751 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-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 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-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19231 |
| This theorem is referenced by: algextdeglem4 33761 aks6d1c6lem5 42178 |
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