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Mirrors > Home > MPE Home > Th. List > prdsplusgcl | Structured version Visualization version GIF version |
Description: Structure product pointwise sums are closed when the factors are monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsplusgcl.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsplusgcl.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsplusgcl.p | ⊢ + = (+g‘𝑌) |
prdsplusgcl.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsplusgcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsplusgcl.r | ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
prdsplusgcl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
prdsplusgcl.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
prdsplusgcl | ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsplusgcl.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdsplusgcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑌) | |
3 | prdsplusgcl.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsplusgcl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | prdsplusgcl.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) | |
6 | 5 | ffnd 6666 | . . 3 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
7 | prdsplusgcl.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
8 | prdsplusgcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
9 | prdsplusgcl.p | . . 3 ⊢ + = (+g‘𝑌) | |
10 | 1, 2, 3, 4, 6, 7, 8, 9 | prdsplusgval 17314 | . 2 ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
11 | 5 | ffvelcdmda 7031 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ Mnd) |
12 | 3 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉) |
13 | 4 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
14 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 Fn 𝐼) |
15 | 7 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝐵) |
16 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | |
17 | 1, 2, 12, 13, 14, 15, 16 | prdsbasprj 17313 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘(𝑅‘𝑥))) |
18 | 8 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ 𝐵) |
19 | 1, 2, 12, 13, 14, 18, 16 | prdsbasprj 17313 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ (Base‘(𝑅‘𝑥))) |
20 | eqid 2737 | . . . . . 6 ⊢ (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝑥)) | |
21 | eqid 2737 | . . . . . 6 ⊢ (+g‘(𝑅‘𝑥)) = (+g‘(𝑅‘𝑥)) | |
22 | 20, 21 | mndcl 18523 | . . . . 5 ⊢ (((𝑅‘𝑥) ∈ Mnd ∧ (𝐹‘𝑥) ∈ (Base‘(𝑅‘𝑥)) ∧ (𝐺‘𝑥) ∈ (Base‘(𝑅‘𝑥))) → ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
23 | 11, 17, 19, 22 | syl3anc 1371 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
24 | 23 | ralrimiva 3141 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
25 | 1, 2, 3, 4, 6 | prdsbasmpt 17311 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥)))) |
26 | 24, 25 | mpbird 256 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) ∈ 𝐵) |
27 | 10, 26 | eqeltrd 2838 | 1 ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ↦ cmpt 5186 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 +gcplusg 17092 Xscprds 17286 Mndcmnd 18515 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-sup 9336 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-fz 13379 df-struct 16978 df-slot 17013 df-ndx 17025 df-base 17043 df-plusg 17105 df-mulr 17106 df-sca 17108 df-vsca 17109 df-ip 17110 df-tset 17111 df-ple 17112 df-ds 17114 df-hom 17116 df-cco 17117 df-prds 17288 df-mgm 18456 df-sgrp 18505 df-mnd 18516 |
This theorem is referenced by: prdsmndd 18548 prdsringd 19988 dsmmacl 21099 |
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