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Mirrors > Home > MPE Home > Th. List > prds1 | Structured version Visualization version GIF version |
Description: Value of the ring unity in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.) |
Ref | Expression |
---|---|
prds1.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prds1.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prds1.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prds1.r | ⊢ (𝜑 → 𝑅:𝐼⟶Ring) |
Ref | Expression |
---|---|
prds1 | ⊢ (𝜑 → (1r ∘ 𝑅) = (1r‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . . . 4 ⊢ (𝑆Xs(mulGrp ∘ 𝑅)) = (𝑆Xs(mulGrp ∘ 𝑅)) | |
2 | prds1.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
3 | prds1.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | mgpf 20029 | . . . . 5 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd | |
5 | prds1.r | . . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶Ring) | |
6 | fco2 6731 | . . . . 5 ⊢ (((mulGrp ↾ Ring):Ring⟶Mnd ∧ 𝑅:𝐼⟶Ring) → (mulGrp ∘ 𝑅):𝐼⟶Mnd) | |
7 | 4, 5, 6 | sylancr 587 | . . . 4 ⊢ (𝜑 → (mulGrp ∘ 𝑅):𝐼⟶Mnd) |
8 | 1, 2, 3, 7 | prds0g 18636 | . . 3 ⊢ (𝜑 → (0g ∘ (mulGrp ∘ 𝑅)) = (0g‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
9 | eqidd 2732 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(mulGrp‘𝑌))) | |
10 | prds1.y | . . . . . 6 ⊢ 𝑌 = (𝑆Xs𝑅) | |
11 | eqid 2731 | . . . . . 6 ⊢ (mulGrp‘𝑌) = (mulGrp‘𝑌) | |
12 | 5 | ffnd 6705 | . . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼) |
13 | 10, 11, 1, 2, 3, 12 | prdsmgp 20087 | . . . . 5 ⊢ (𝜑 → ((Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅))) ∧ (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅))))) |
14 | 13 | simpld 495 | . . . 4 ⊢ (𝜑 → (Base‘(mulGrp‘𝑌)) = (Base‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
15 | 13 | simprd 496 | . . . . 5 ⊢ (𝜑 → (+g‘(mulGrp‘𝑌)) = (+g‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
16 | 15 | oveqdr 7421 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑌)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑌)))) → (𝑥(+g‘(mulGrp‘𝑌))𝑦) = (𝑥(+g‘(𝑆Xs(mulGrp ∘ 𝑅)))𝑦)) |
17 | 9, 14, 16 | grpidpropd 18563 | . . 3 ⊢ (𝜑 → (0g‘(mulGrp‘𝑌)) = (0g‘(𝑆Xs(mulGrp ∘ 𝑅)))) |
18 | 8, 17 | eqtr4d 2774 | . 2 ⊢ (𝜑 → (0g ∘ (mulGrp ∘ 𝑅)) = (0g‘(mulGrp‘𝑌))) |
19 | df-ur 19964 | . . . 4 ⊢ 1r = (0g ∘ mulGrp) | |
20 | 19 | coeq1i 5851 | . . 3 ⊢ (1r ∘ 𝑅) = ((0g ∘ mulGrp) ∘ 𝑅) |
21 | coass 6253 | . . 3 ⊢ ((0g ∘ mulGrp) ∘ 𝑅) = (0g ∘ (mulGrp ∘ 𝑅)) | |
22 | 20, 21 | eqtri 2759 | . 2 ⊢ (1r ∘ 𝑅) = (0g ∘ (mulGrp ∘ 𝑅)) |
23 | eqid 2731 | . . 3 ⊢ (1r‘𝑌) = (1r‘𝑌) | |
24 | 11, 23 | ringidval 19965 | . 2 ⊢ (1r‘𝑌) = (0g‘(mulGrp‘𝑌)) |
25 | 18, 22, 24 | 3eqtr4g 2796 | 1 ⊢ (𝜑 → (1r ∘ 𝑅) = (1r‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ↾ cres 5671 ∘ ccom 5673 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 0gc0g 17367 Xscprds 17373 Mndcmnd 18602 mulGrpcmgp 19946 1rcur 19963 Ringcrg 20014 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-sup 9419 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-hom 17203 df-cco 17204 df-0g 17369 df-prds 17375 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-mgp 19947 df-ur 19964 df-ring 20016 |
This theorem is referenced by: pws1 20093 |
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