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| Mirrors > Home > MPE Home > Th. List > cycsubmcmn | Structured version Visualization version GIF version | ||
| Description: The set of nonnegative integer powers of an element 𝐴 of a monoid forms a commutative monoid. (Contributed by AV, 20-Jan-2024.) | 
| Ref | Expression | 
|---|---|
| cycsubmcmn.b | ⊢ 𝐵 = (Base‘𝐺) | 
| cycsubmcmn.t | ⊢ · = (.g‘𝐺) | 
| cycsubmcmn.f | ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | 
| cycsubmcmn.c | ⊢ 𝐶 = ran 𝐹 | 
| Ref | Expression | 
|---|---|
| cycsubmcmn | ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s 𝐶) ∈ CMnd) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | cycsubmcmn.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | cycsubmcmn.t | . . . 4 ⊢ · = (.g‘𝐺) | |
| 3 | cycsubmcmn.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | |
| 4 | cycsubmcmn.c | . . . 4 ⊢ 𝐶 = ran 𝐹 | |
| 5 | 1, 2, 3, 4 | cycsubm 19220 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → 𝐶 ∈ (SubMnd‘𝐺)) | 
| 6 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 7 | eqid 2737 | . . . . . 6 ⊢ (𝐺 ↾s 𝐶) = (𝐺 ↾s 𝐶) | |
| 8 | 1, 6, 7 | issubm2 18817 | . . . . 5 ⊢ (𝐺 ∈ Mnd → (𝐶 ∈ (SubMnd‘𝐺) ↔ (𝐶 ⊆ 𝐵 ∧ (0g‘𝐺) ∈ 𝐶 ∧ (𝐺 ↾s 𝐶) ∈ Mnd))) | 
| 9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐶 ∈ (SubMnd‘𝐺) ↔ (𝐶 ⊆ 𝐵 ∧ (0g‘𝐺) ∈ 𝐶 ∧ (𝐺 ↾s 𝐶) ∈ Mnd))) | 
| 10 | simp3 1139 | . . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ (0g‘𝐺) ∈ 𝐶 ∧ (𝐺 ↾s 𝐶) ∈ Mnd) → (𝐺 ↾s 𝐶) ∈ Mnd) | |
| 11 | 9, 10 | biimtrdi 253 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐶 ∈ (SubMnd‘𝐺) → (𝐺 ↾s 𝐶) ∈ Mnd)) | 
| 12 | 5, 11 | mpd 15 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s 𝐶) ∈ Mnd) | 
| 13 | 7 | submbas 18827 | . . . . . . . 8 ⊢ (𝐶 ∈ (SubMnd‘𝐺) → 𝐶 = (Base‘(𝐺 ↾s 𝐶))) | 
| 14 | 5, 13 | syl 17 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → 𝐶 = (Base‘(𝐺 ↾s 𝐶))) | 
| 15 | 14 | eqcomd 2743 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (Base‘(𝐺 ↾s 𝐶)) = 𝐶) | 
| 16 | 15 | eleq2d 2827 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ (Base‘(𝐺 ↾s 𝐶)) ↔ 𝑥 ∈ 𝐶)) | 
| 17 | 15 | eleq2d 2827 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝑦 ∈ (Base‘(𝐺 ↾s 𝐶)) ↔ 𝑦 ∈ 𝐶)) | 
| 18 | 16, 17 | anbi12d 632 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ (Base‘(𝐺 ↾s 𝐶)) ∧ 𝑦 ∈ (Base‘(𝐺 ↾s 𝐶))) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶))) | 
| 19 | eqid 2737 | . . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 20 | 1, 2, 3, 4, 19 | cycsubmcom 19222 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)) | 
| 21 | 5 | adantr 480 | . . . . . . 7 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → 𝐶 ∈ (SubMnd‘𝐺)) | 
| 22 | 7, 19 | ressplusg 17334 | . . . . . . . . . 10 ⊢ (𝐶 ∈ (SubMnd‘𝐺) → (+g‘𝐺) = (+g‘(𝐺 ↾s 𝐶))) | 
| 23 | 22 | eqcomd 2743 | . . . . . . . . 9 ⊢ (𝐶 ∈ (SubMnd‘𝐺) → (+g‘(𝐺 ↾s 𝐶)) = (+g‘𝐺)) | 
| 24 | 23 | oveqd 7448 | . . . . . . . 8 ⊢ (𝐶 ∈ (SubMnd‘𝐺) → (𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑥(+g‘𝐺)𝑦)) | 
| 25 | 23 | oveqd 7448 | . . . . . . . 8 ⊢ (𝐶 ∈ (SubMnd‘𝐺) → (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥) = (𝑦(+g‘𝐺)𝑥)) | 
| 26 | 24, 25 | eqeq12d 2753 | . . . . . . 7 ⊢ (𝐶 ∈ (SubMnd‘𝐺) → ((𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) | 
| 27 | 21, 26 | syl 17 | . . . . . 6 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → ((𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) | 
| 28 | 20, 27 | mpbird 257 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥)) | 
| 29 | 28 | ex 412 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥))) | 
| 30 | 18, 29 | sylbid 240 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ (Base‘(𝐺 ↾s 𝐶)) ∧ 𝑦 ∈ (Base‘(𝐺 ↾s 𝐶))) → (𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥))) | 
| 31 | 30 | ralrimivv 3200 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ∀𝑥 ∈ (Base‘(𝐺 ↾s 𝐶))∀𝑦 ∈ (Base‘(𝐺 ↾s 𝐶))(𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥)) | 
| 32 | eqid 2737 | . . 3 ⊢ (Base‘(𝐺 ↾s 𝐶)) = (Base‘(𝐺 ↾s 𝐶)) | |
| 33 | eqid 2737 | . . 3 ⊢ (+g‘(𝐺 ↾s 𝐶)) = (+g‘(𝐺 ↾s 𝐶)) | |
| 34 | 32, 33 | iscmn 19807 | . 2 ⊢ ((𝐺 ↾s 𝐶) ∈ CMnd ↔ ((𝐺 ↾s 𝐶) ∈ Mnd ∧ ∀𝑥 ∈ (Base‘(𝐺 ↾s 𝐶))∀𝑦 ∈ (Base‘(𝐺 ↾s 𝐶))(𝑥(+g‘(𝐺 ↾s 𝐶))𝑦) = (𝑦(+g‘(𝐺 ↾s 𝐶))𝑥))) | 
| 35 | 12, 31, 34 | sylanbrc 583 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐺 ↾s 𝐶) ∈ CMnd) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 ↦ cmpt 5225 ran crn 5686 ‘cfv 6561 (class class class)co 7431 ℕ0cn0 12526 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 SubMndcsubmnd 18795 .gcmg 19085 CMndccmn 19798 | 
| 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-submnd 18797 df-mulg 19086 df-cmn 19800 | 
| This theorem is referenced by: (None) | 
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