Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cntzmhm2 | Structured version Visualization version GIF version |
Description: Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.) |
Ref | Expression |
---|---|
cntzmhm.z | ⊢ 𝑍 = (Cntz‘𝐺) |
cntzmhm.y | ⊢ 𝑌 = (Cntz‘𝐻) |
Ref | Expression |
---|---|
cntzmhm2 | ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntzmhm.z | . . . . 5 ⊢ 𝑍 = (Cntz‘𝐺) | |
2 | cntzmhm.y | . . . . 5 ⊢ 𝑌 = (Cntz‘𝐻) | |
3 | 1, 2 | cntzmhm 18407 | . . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍‘𝑇)) → (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))) |
4 | 3 | ralrimiva 3179 | . . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍‘𝑇)(𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))) |
5 | ssralv 4030 | . . 3 ⊢ (𝑆 ⊆ (𝑍‘𝑇) → (∀𝑥 ∈ (𝑍‘𝑇)(𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))) | |
6 | 4, 5 | mpan9 507 | . 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))) |
7 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
8 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
9 | 7, 8 | mhmf 17949 | . . . . 5 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻)) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻)) |
11 | 10 | ffund 6511 | . . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → Fun 𝐹) |
12 | simpr 485 | . . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ (𝑍‘𝑇)) | |
13 | 7, 1 | cntzssv 18396 | . . . . 5 ⊢ (𝑍‘𝑇) ⊆ (Base‘𝐺) |
14 | 12, 13 | sstrdi 3976 | . . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ (Base‘𝐺)) |
15 | 10 | fdmd 6516 | . . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → dom 𝐹 = (Base‘𝐺)) |
16 | 14, 15 | sseqtrrd 4005 | . . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ dom 𝐹) |
17 | funimass4 6723 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → ((𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))) | |
18 | 11, 16, 17 | syl2anc 584 | . 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → ((𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))) |
19 | 6, 18 | mpbird 258 | 1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 dom cdm 5548 “ cima 5551 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 MndHom cmhm 17942 Cntzccntz 18383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-mhm 17944 df-cntz 18385 |
This theorem is referenced by: gsumzmhm 18986 gsumzinv 18994 |
Copyright terms: Public domain | W3C validator |