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| Mirrors > Home > MPE Home > Th. List > Mathboxes > subgmulgcld | Structured version Visualization version GIF version | ||
| Description: Closure of the group multiple within a subgroup. (Contributed by Thierry Arnoux, 5-Oct-2025.) |
| Ref | Expression |
|---|---|
| subgmulgcld.b | ⊢ 𝐵 = (Base‘𝑅) |
| subgmulgcld.x | ⊢ · = (.g‘𝑅) |
| subgmulgcld.r | ⊢ (𝜑 → 𝑅 ∈ Grp) |
| subgmulgcld.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
| subgmulgcld.s | ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝑅)) |
| subgmulgcld.z | ⊢ (𝜑 → 𝑍 ∈ ℤ) |
| Ref | Expression |
|---|---|
| subgmulgcld | ⊢ (𝜑 → (𝑍 · 𝐴) ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘(𝑅 ↾s 𝑆)) = (Base‘(𝑅 ↾s 𝑆)) | |
| 2 | eqid 2737 | . . 3 ⊢ (.g‘(𝑅 ↾s 𝑆)) = (.g‘(𝑅 ↾s 𝑆)) | |
| 3 | subgmulgcld.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝑅)) | |
| 4 | eqid 2737 | . . . . 5 ⊢ (𝑅 ↾s 𝑆) = (𝑅 ↾s 𝑆) | |
| 5 | 4 | subggrp 19147 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (𝑅 ↾s 𝑆) ∈ Grp) |
| 6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → (𝑅 ↾s 𝑆) ∈ Grp) |
| 7 | subgmulgcld.z | . . 3 ⊢ (𝜑 → 𝑍 ∈ ℤ) | |
| 8 | subgmulgcld.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 9 | subgmulgcld.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 10 | 9 | subgss 19145 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → 𝑆 ⊆ 𝐵) |
| 11 | 4, 9 | ressbas2 17283 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘(𝑅 ↾s 𝑆))) |
| 12 | 3, 10, 11 | 3syl 18 | . . . 4 ⊢ (𝜑 → 𝑆 = (Base‘(𝑅 ↾s 𝑆))) |
| 13 | 8, 12 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘(𝑅 ↾s 𝑆))) |
| 14 | 1, 2, 6, 7, 13 | mulgcld 19114 | . 2 ⊢ (𝜑 → (𝑍(.g‘(𝑅 ↾s 𝑆))𝐴) ∈ (Base‘(𝑅 ↾s 𝑆))) |
| 15 | subgmulgcld.x | . . . 4 ⊢ · = (.g‘𝑅) | |
| 16 | 15, 4, 2 | subgmulg 19158 | . . 3 ⊢ ((𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑍 ∈ ℤ ∧ 𝐴 ∈ 𝑆) → (𝑍 · 𝐴) = (𝑍(.g‘(𝑅 ↾s 𝑆))𝐴)) |
| 17 | 3, 7, 8, 16 | syl3anc 1373 | . 2 ⊢ (𝜑 → (𝑍 · 𝐴) = (𝑍(.g‘(𝑅 ↾s 𝑆))𝐴)) |
| 18 | 14, 17, 12 | 3eltr4d 2856 | 1 ⊢ (𝜑 → (𝑍 · 𝐴) ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 ℤcz 12613 Basecbs 17247 ↾s cress 17274 Grpcgrp 18951 .gcmg 19085 SubGrpcsubg 19138 |
| 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-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-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-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 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-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-mulg 19086 df-subg 19141 |
| This theorem is referenced by: elrgspnlem4 33249 elrgspnsubrunlem2 33252 |
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