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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ofaddmndmap | Structured version Visualization version GIF version |
Description: The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.) |
Ref | Expression |
---|---|
ofaddmndmap.r | ⊢ 𝑅 = (Base‘𝑀) |
ofaddmndmap.p | ⊢ + = (+g‘𝑀) |
Ref | Expression |
---|---|
ofaddmndmap | ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1188 | . . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑀 ∈ Mnd) | |
2 | simprl 769 | . . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅) | |
3 | simprr 771 | . . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅) | |
4 | ofaddmndmap.r | . . . . 5 ⊢ 𝑅 = (Base‘𝑀) | |
5 | ofaddmndmap.p | . . . . 5 ⊢ + = (+g‘𝑀) | |
6 | 4, 5 | mndcl 18711 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥 + 𝑦) ∈ 𝑅) |
7 | 1, 2, 3, 6 | syl3anc 1368 | . . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥 + 𝑦) ∈ 𝑅) |
8 | elmapi 8876 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑m 𝑉) → 𝐴:𝑉⟶𝑅) | |
9 | 8 | adantr 479 | . . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐴:𝑉⟶𝑅) |
10 | 9 | 3ad2ant3 1132 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐴:𝑉⟶𝑅) |
11 | elmapi 8876 | . . . . 5 ⊢ (𝐵 ∈ (𝑅 ↑m 𝑉) → 𝐵:𝑉⟶𝑅) | |
12 | 11 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉)) → 𝐵:𝑉⟶𝑅) |
13 | 12 | 3ad2ant3 1132 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝐵:𝑉⟶𝑅) |
14 | simp2 1134 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → 𝑉 ∈ 𝑌) | |
15 | inidm 4221 | . . 3 ⊢ (𝑉 ∩ 𝑉) = 𝑉 | |
16 | 7, 10, 13, 14, 14, 15 | off 7710 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵):𝑉⟶𝑅) |
17 | 4 | fvexi 6916 | . . 3 ⊢ 𝑅 ∈ V |
18 | elmapg 8866 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝑌) → ((𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉) ↔ (𝐴 ∘f + 𝐵):𝑉⟶𝑅)) | |
19 | 17, 14, 18 | sylancr 585 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → ((𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉) ↔ (𝐴 ∘f + 𝐵):𝑉⟶𝑅)) |
20 | 16, 19 | mpbird 256 | 1 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∘f cof 7690 ↑m cmap 8853 Basecbs 17189 +gcplusg 17242 Mndcmnd 18703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-1st 8001 df-2nd 8002 df-map 8855 df-mgm 18609 df-sgrp 18688 df-mnd 18704 |
This theorem is referenced by: lincsumcl 47595 |
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