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Mirrors > Home > MPE Home > Th. List > cycsubmcom | Structured version Visualization version GIF version |
Description: The operation of a monoid is commutative over the set of nonnegative integer powers of an element 𝐴 of the monoid. (Contributed by AV, 20-Jan-2024.) |
Ref | Expression |
---|---|
cycsubmcom.b | ⊢ 𝐵 = (Base‘𝐺) |
cycsubmcom.t | ⊢ · = (.g‘𝐺) |
cycsubmcom.f | ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) |
cycsubmcom.c | ⊢ 𝐶 = ran 𝐹 |
cycsubmcom.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
cycsubmcom | ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycsubmcom.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
2 | cycsubmcom.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
3 | cycsubmcom.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ ℕ0 ↦ (𝑥 · 𝐴)) | |
4 | cycsubmcom.c | . . . . . 6 ⊢ 𝐶 = ran 𝐹 | |
5 | 1, 2, 3, 4 | cycsubmel 18339 | . . . . 5 ⊢ (𝑐 ∈ 𝐶 ↔ ∃𝑖 ∈ ℕ0 𝑐 = (𝑖 · 𝐴)) |
6 | 5 | biimpi 218 | . . . 4 ⊢ (𝑐 ∈ 𝐶 → ∃𝑖 ∈ ℕ0 𝑐 = (𝑖 · 𝐴)) |
7 | 6 | adantl 484 | . . 3 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ 𝑐 ∈ 𝐶) → ∃𝑖 ∈ ℕ0 𝑐 = (𝑖 · 𝐴)) |
8 | 7 | ralrimiva 3181 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → ∀𝑐 ∈ 𝐶 ∃𝑖 ∈ ℕ0 𝑐 = (𝑖 · 𝐴)) |
9 | simplll 773 | . . . 4 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) → 𝐺 ∈ Mnd) | |
10 | simprl 769 | . . . 4 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) → 𝑚 ∈ ℕ0) | |
11 | simprr 771 | . . . 4 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0) | |
12 | simpllr 774 | . . . 4 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) → 𝐴 ∈ 𝐵) | |
13 | cycsubmcom.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
14 | 1, 2, 13 | mulgnn0dir 18253 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 𝐴 ∈ 𝐵)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴))) |
15 | 9, 10, 11, 12, 14 | syl13anc 1367 | . . 3 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) ∧ (𝑚 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0)) → ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴))) |
16 | 15 | ralrimivva 3190 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → ∀𝑚 ∈ ℕ0 ∀𝑛 ∈ ℕ0 ((𝑚 + 𝑛) · 𝐴) = ((𝑚 · 𝐴) + (𝑛 · 𝐴))) |
17 | simprl 769 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑋 ∈ 𝐶) | |
18 | simprr 771 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → 𝑌 ∈ 𝐶) | |
19 | nn0sscn 11900 | . . 3 ⊢ ℕ0 ⊆ ℂ | |
20 | 19 | a1i 11 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → ℕ0 ⊆ ℂ) |
21 | 8, 16, 17, 18, 20 | cyccom 18342 | 1 ⊢ (((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝐵) ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3138 ⊆ wss 3933 ↦ cmpt 5143 ran crn 5553 ‘cfv 6352 (class class class)co 7153 ℂcc 10532 + caddc 10537 ℕ0cn0 11895 Basecbs 16479 +gcplusg 16561 Mndcmnd 17907 .gcmg 18220 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-1st 7686 df-2nd 7687 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-n0 11896 df-z 11980 df-uz 12242 df-fz 12891 df-seq 13368 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-mulg 18221 |
This theorem is referenced by: cycsubmcmn 19004 |
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