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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 6515 | . . 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 16746 | . 2 ⊢ (𝜑 → (𝐹 + 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
11 | 5 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ Mnd) |
12 | 3 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑆 ∈ 𝑉) |
13 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊) |
14 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 Fn 𝐼) |
15 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐹 ∈ 𝐵) |
16 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) | |
17 | 1, 2, 12, 13, 14, 15, 16 | prdsbasprj 16745 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ (Base‘(𝑅‘𝑥))) |
18 | 8 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ 𝐵) |
19 | 1, 2, 12, 13, 14, 18, 16 | prdsbasprj 16745 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ (Base‘(𝑅‘𝑥))) |
20 | eqid 2821 | . . . . . 6 ⊢ (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝑥)) | |
21 | eqid 2821 | . . . . . 6 ⊢ (+g‘(𝑅‘𝑥)) = (+g‘(𝑅‘𝑥)) | |
22 | 20, 21 | mndcl 17919 | . . . . 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 1367 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
24 | 23 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
25 | 1, 2, 3, 4, 6 | prdsbasmpt 16743 | . . 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 259 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(+g‘(𝑅‘𝑥))(𝐺‘𝑥))) ∈ 𝐵) |
27 | 10, 26 | eqeltrd 2913 | 1 ⊢ (𝜑 → (𝐹 + 𝐺) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ↦ cmpt 5146 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Xscprds 16719 Mndcmnd 17911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-hom 16589 df-cco 16590 df-prds 16721 df-mgm 17852 df-sgrp 17901 df-mnd 17912 |
This theorem is referenced by: prdsmndd 17944 prdsringd 19362 dsmmacl 20885 |
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