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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 1134 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑅 ∈ Mnd) | 
| 8 | simp2 1138 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ 𝑉) | |
| 9 | 2 | 3ad2ant1 1134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑉 = (Base‘𝑅)) | 
| 10 | 8, 9 | eleqtrd 2843 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) | 
| 11 | simp3 1139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ 𝑉) | |
| 12 | 11, 9 | eleqtrd 2843 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → 𝑦 ∈ (Base‘𝑅)) | 
| 13 | eqid 2737 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 14 | 13, 3 | mndcl 18755 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥 + 𝑦) ∈ (Base‘𝑅)) | 
| 15 | 7, 10, 12, 14 | syl3anc 1373 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ (Base‘𝑅)) | 
| 16 | 15, 9 | eleqtrrd 2844 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑥 + 𝑦) ∈ 𝑉) | 
| 17 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑅 ∈ Mnd) | 
| 18 | 10 | 3adant3r3 1185 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑥 ∈ (Base‘𝑅)) | 
| 19 | 12 | 3adant3r3 1185 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑦 ∈ (Base‘𝑅)) | 
| 20 | simpr3 1197 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ 𝑉) | |
| 21 | 2 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) | 
| 22 | 20, 21 | eleqtrd 2843 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → 𝑧 ∈ (Base‘𝑅)) | 
| 23 | 13, 3 | mndass 18756 | . . . 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 1374 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | 
| 25 | 24 | fveq2d 6910 | . 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 18762 | . . . 4 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅)) | 
| 28 | 6, 27 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ (Base‘𝑅)) | 
| 29 | 28, 2 | eleqtrrd 2844 | . 2 ⊢ (𝜑 → 0 ∈ 𝑉) | 
| 30 | 2 | eleq2d 2827 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↔ 𝑥 ∈ (Base‘𝑅))) | 
| 31 | 30 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (Base‘𝑅)) | 
| 32 | 13, 3, 26 | mndlid 18767 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → ( 0 + 𝑥) = 𝑥) | 
| 33 | 6, 31, 32 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ( 0 + 𝑥) = 𝑥) | 
| 34 | 33 | fveq2d 6910 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘( 0 + 𝑥)) = (𝐹‘𝑥)) | 
| 35 | 13, 3, 26 | mndrid 18768 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 + 0 ) = 𝑥) | 
| 36 | 6, 31, 35 | syl2an2r 685 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝑥 + 0 ) = 𝑥) | 
| 37 | 36 | fveq2d 6910 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝐹‘(𝑥 + 0 )) = (𝐹‘𝑥)) | 
| 38 | 1, 2, 3, 4, 5, 6, 16, 25, 29, 34, 37 | imasmnd2 18787 | 1 ⊢ (𝜑 → (𝑈 ∈ Mnd ∧ (𝐹‘ 0 ) = (0g‘𝑈))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 –onto→wfo 6559 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 “s cimas 17549 Mndcmnd 18747 | 
| 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-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-slot 17219 df-ndx 17231 df-base 17248 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-mgm 18653 df-sgrp 18732 df-mnd 18748 | 
| This theorem is referenced by: imasmndf1 18789 imasmhm 33382 r1pquslmic 33631 | 
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