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 1064 | . . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑀 ∈ Mnd) | |
2 | simprl 794 | . . . 4 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅) | |
3 | simprr 796 | . . . 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 17301 | . . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅) → (𝑥 + 𝑦) ∈ 𝑅) |
7 | 1, 2, 3, 6 | syl3anc 1326 | . . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥 + 𝑦) ∈ 𝑅) |
8 | elmapi 7879 | . . . . 5 ⊢ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) → 𝐴:𝑉⟶𝑅) | |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉)) → 𝐴:𝑉⟶𝑅) |
10 | 9 | 3ad2ant3 1084 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝐴:𝑉⟶𝑅) |
11 | elmapi 7879 | . . . . 5 ⊢ (𝐵 ∈ (𝑅 ↑𝑚 𝑉) → 𝐵:𝑉⟶𝑅) | |
12 | 11 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉)) → 𝐵:𝑉⟶𝑅) |
13 | 12 | 3ad2ant3 1084 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝐵:𝑉⟶𝑅) |
14 | simp2 1062 | . . 3 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → 𝑉 ∈ 𝑌) | |
15 | inidm 3822 | . . 3 ⊢ (𝑉 ∩ 𝑉) = 𝑉 | |
16 | 7, 10, 13, 14, 14, 15 | off 6912 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → (𝐴 ∘𝑓 + 𝐵):𝑉⟶𝑅) |
17 | fvex 6201 | . . . 4 ⊢ (Base‘𝑀) ∈ V | |
18 | 4, 17 | eqeltri 2697 | . . 3 ⊢ 𝑅 ∈ V |
19 | elmapg 7870 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝑉 ∈ 𝑌) → ((𝐴 ∘𝑓 + 𝐵) ∈ (𝑅 ↑𝑚 𝑉) ↔ (𝐴 ∘𝑓 + 𝐵):𝑉⟶𝑅)) | |
20 | 18, 14, 19 | sylancr 695 | . 2 ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑𝑚 𝑉) ∧ 𝐵 ∈ (𝑅 ↑𝑚 𝑉))) → ((𝐴 ∘𝑓 + 𝐵) ∈ (𝑅 ↑𝑚 𝑉) ↔ (𝐴 ∘𝑓 + 𝐵):𝑉⟶𝑅)) |
21 | 16, 20 | mpbird 247 | 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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ↑𝑚 cmap 7857 Basecbs 15857 +gcplusg 15941 Mndcmnd 17294 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-1st 7168 df-2nd 7169 df-map 7859 df-mgm 17242 df-sgrp 17284 df-mnd 17295 |
This theorem is referenced by: lincsumcl 42220 |
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