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Mirrors > Home > MPE Home > Th. List > imasmnd | Structured version Visualization version GIF version |
Description: The image structure of a monoid is a monoid. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
imasmnd.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasmnd.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasmnd.p | ⊢ + = (+g‘𝑅) |
imasmnd.f | ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
imasmnd.e | ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) |
imasmnd.r | ⊢ (𝜑 → 𝑅 ∈ Mnd) |
imasmnd.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
imasmnd | ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasmnd.u | . 2 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imasmnd.v | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imasmnd.p | . 2 ⊢ + = (+g‘𝑅) | |
4 | imasmnd.f | . 2 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | |
5 | imasmnd.e | . 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞)))) | |
6 | imasmnd.r | . 2 ⊢ (𝜑 → 𝑅 ∈ Mnd) | |
7 | 6 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑅 ∈ Mnd) |
8 | simp2 1062 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉) | |
9 | 2 | 3ad2ant1 1082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅)) |
10 | 8, 9 | eleqtrd 2703 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) |
11 | simp3 1063 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉) | |
12 | 11, 9 | eleqtrd 2703 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅)) |
13 | eqid 2622 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13, 3 | mndcl 17301 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅)) |
15 | 7, 10, 12, 14 | syl3anc 1326 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ (Base‘𝑅)) |
16 | 15, 9 | eleqtrrd 2704 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) |
17 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Mnd) |
18 | 10 | 3adant3r3 1276 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅)) |
19 | 12 | 3adant3r3 1276 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅)) |
20 | simpr3 1069 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉) | |
21 | 2 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) |
22 | 20, 21 | eleqtrd 2703 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅)) |
23 | 13, 3 | mndass 17302 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
24 | 17, 18, 19, 22, 23 | syl13anc 1328 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
25 | 24 | fveq2d 6195 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → (𝐹‘((𝑥 + 𝑦) + 𝑧)) = (𝐹‘(𝑥 + (𝑦 + 𝑧)))) |
26 | imasmnd.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
27 | 13, 26 | mndidcl 17308 | . . . 4 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
28 | 6, 27 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
29 | 28, 2 | eleqtrrd 2704 | . 2 ⊢ (𝜑 → 0 ∈ 𝑉) |
30 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑅 ∈ Mnd) |
31 | 2 | eleq2d 2687 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅))) |
32 | 31 | biimpa 501 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) |
33 | 13, 3, 26 | mndlid 17311 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 + 𝑥) = 𝑥) |
34 | 30, 32, 33 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = 𝑥) |
35 | 34 | fveq2d 6195 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥)) |
36 | 13, 3, 26 | mndrid 17312 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 + 0 ) = 𝑥) |
37 | 30, 32, 36 | syl2anc 693 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 + 0 ) = 𝑥) |
38 | 37 | fveq2d 6195 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹‘𝑥)) |
39 | 1, 2, 3, 4, 5, 6, 16, 25, 29, 35, 38 | imasmnd2 17327 | 1 ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 “s cimas 16164 Mndcmnd 17294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-imas 16168 df-mgm 17242 df-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: imasmndf1 17329 |
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