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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 1125 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑅 ∈ Mnd) |
8 | simp2 1129 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉) | |
9 | 2 | 3ad2ant1 1125 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅)) |
10 | 8, 9 | eleqtrd 2912 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) |
11 | simp3 1130 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉) | |
12 | 11, 9 | eleqtrd 2912 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅)) |
13 | eqid 2818 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
14 | 13, 3 | mndcl 17907 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅)) |
15 | 7, 10, 12, 14 | syl3anc 1363 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ (Base‘𝑅)) |
16 | 15, 9 | eleqtrrd 2913 | . 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 1176 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅)) |
19 | 12 | 3adant3r3 1176 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅)) |
20 | simpr3 1188 | . . . . 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 2912 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅)) |
23 | 13, 3 | mndass 17908 | . . . 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 1364 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
25 | 24 | fveq2d 6667 | . 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 17914 | . . . 4 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) |
28 | 6, 27 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) |
29 | 28, 2 | eleqtrrd 2913 | . 2 ⊢ (𝜑 → 0 ∈ 𝑉) |
30 | 2 | eleq2d 2895 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅))) |
31 | 30 | biimpa 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) |
32 | 13, 3, 26 | mndlid 17919 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 + 𝑥) = 𝑥) |
33 | 6, 31, 32 | syl2an2r 681 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = 𝑥) |
34 | 33 | fveq2d 6667 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥)) |
35 | 13, 3, 26 | mndrid 17920 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 + 0 ) = 𝑥) |
36 | 6, 31, 35 | syl2an2r 681 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 + 0 ) = 𝑥) |
37 | 36 | fveq2d 6667 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹‘𝑥)) |
38 | 1, 2, 3, 4, 5, 6, 16, 25, 29, 34, 37 | imasmnd2 17936 | 1 ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 0gc0g 16701 “s cimas 16765 Mndcmnd 17899 |
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 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 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-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 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-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-0g 16703 df-imas 16769 df-mgm 17840 df-sgrp 17889 df-mnd 17900 |
This theorem is referenced by: imasmndf1 17938 |
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