Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > prdsmulrcl | Structured version Visualization version GIF version |
Description: A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
prdsmulrcl.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsmulrcl.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsmulrcl.t | ⊢ · = (.r‘𝑌) |
prdsmulrcl.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsmulrcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsmulrcl.r | ⊢ (𝜑 → 𝑅:𝐼⟶Ring) |
prdsmulrcl.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
prdsmulrcl.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
Ref | Expression |
---|---|
prdsmulrcl | ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsmulrcl.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdsmulrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑌) | |
3 | prdsmulrcl.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | prdsmulrcl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | prdsmulrcl.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Ring) | |
6 | 5 | ffnd 6508 | . . 3 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
7 | prdsmulrcl.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
8 | prdsmulrcl.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
9 | prdsmulrcl.t | . . 3 ⊢ · = (.r‘𝑌) | |
10 | 1, 2, 3, 4, 6, 7, 8, 9 | prdsmulrval 16736 | . 2 ⊢ (𝜑 → (𝐹 · 𝐺) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
11 | 5 | ffvelrnda 6843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ Ring) |
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 16733 | . . . . 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 16733 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ (Base‘(𝑅‘𝑥))) |
20 | eqid 2818 | . . . . . 6 ⊢ (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝑥)) | |
21 | eqid 2818 | . . . . . 6 ⊢ (.r‘(𝑅‘𝑥)) = (.r‘(𝑅‘𝑥)) | |
22 | 20, 21 | ringcl 19240 | . . . . 5 ⊢ (((𝑅‘𝑥) ∈ Ring ∧ (𝐹‘𝑥) ∈ (Base‘(𝑅‘𝑥)) ∧ (𝐺‘𝑥) ∈ (Base‘(𝑅‘𝑥))) → ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
23 | 11, 17, 19, 22 | syl3anc 1363 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
24 | 23 | ralrimiva 3179 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥))) |
25 | 1, 2, 3, 4, 6 | prdsbasmpt 16731 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥))) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐼 ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥)) ∈ (Base‘(𝑅‘𝑥)))) |
26 | 24, 25 | mpbird 258 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(.r‘(𝑅‘𝑥))(𝐺‘𝑥))) ∈ 𝐵) |
27 | 10, 26 | eqeltrd 2910 | 1 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ↦ cmpt 5137 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 .rcmulr 16554 Xscprds 16707 Ringcrg 19226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-hom 16577 df-cco 16578 df-prds 16709 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mgp 19169 df-ring 19228 |
This theorem is referenced by: prdsringd 19291 |
Copyright terms: Public domain | W3C validator |