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Mirrors > Home > MPE Home > Th. List > pws0g | Structured version Visualization version GIF version |
Description: Zero in a structure power of a monoid. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsmnd.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pws0g.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
pws0g | ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 0 }) = (0g‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 ⊢ ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) = ((Scalar‘𝑅)Xs(𝐼 × {𝑅})) | |
2 | simpr 484 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 𝐼 ∈ 𝑉) | |
3 | fvexd 6906 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (Scalar‘𝑅) ∈ V) | |
4 | fconst6g 6780 | . . . 4 ⊢ (𝑅 ∈ Mnd → (𝐼 × {𝑅}):𝐼⟶Mnd) | |
5 | 4 | adantr 480 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (𝐼 × {𝑅}):𝐼⟶Mnd) |
6 | 1, 2, 3, 5 | prds0g 18721 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (0g ∘ (𝐼 × {𝑅})) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
7 | fconstmpt 5734 | . . 3 ⊢ (𝐼 × { 0 }) = (𝑥 ∈ 𝐼 ↦ 0 ) | |
8 | elex 3489 | . . . . 5 ⊢ (𝑅 ∈ Mnd → 𝑅 ∈ V) | |
9 | 8 | ad2antrr 725 | . . . 4 ⊢ (((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ V) |
10 | fconstmpt 5734 | . . . . 5 ⊢ (𝐼 × {𝑅}) = (𝑥 ∈ 𝐼 ↦ 𝑅) | |
11 | 10 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (𝐼 × {𝑅}) = (𝑥 ∈ 𝐼 ↦ 𝑅)) |
12 | fn0g 18616 | . . . . . 6 ⊢ 0g Fn V | |
13 | 12 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 0g Fn V) |
14 | dffn5 6951 | . . . . 5 ⊢ (0g Fn V ↔ 0g = (𝑟 ∈ V ↦ (0g‘𝑟))) | |
15 | 13, 14 | sylib 217 | . . . 4 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 0g = (𝑟 ∈ V ↦ (0g‘𝑟))) |
16 | fveq2 6891 | . . . . 5 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅)) | |
17 | pws0g.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
18 | 16, 17 | eqtr4di 2786 | . . . 4 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 ) |
19 | 9, 11, 15, 18 | fmptco 7132 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (0g ∘ (𝐼 × {𝑅})) = (𝑥 ∈ 𝐼 ↦ 0 )) |
20 | 7, 19 | eqtr4id 2787 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 0 }) = (0g ∘ (𝐼 × {𝑅}))) |
21 | pwsmnd.y | . . . 4 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
22 | eqid 2728 | . . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
23 | 21, 22 | pwsval 17461 | . . 3 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → 𝑌 = ((Scalar‘𝑅)Xs(𝐼 × {𝑅}))) |
24 | 23 | fveq2d 6895 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (0g‘𝑌) = (0g‘((Scalar‘𝑅)Xs(𝐼 × {𝑅})))) |
25 | 6, 20, 24 | 3eqtr4d 2778 | 1 ⊢ ((𝑅 ∈ Mnd ∧ 𝐼 ∈ 𝑉) → (𝐼 × { 0 }) = (0g‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 {csn 4624 ↦ cmpt 5225 × cxp 5670 ∘ ccom 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Scalarcsca 17229 0gc0g 17414 Xscprds 17420 ↑s cpws 17421 Mndcmnd 18687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-hom 17250 df-cco 17251 df-0g 17416 df-prds 17422 df-pws 17424 df-mgm 18593 df-sgrp 18672 df-mnd 18688 |
This theorem is referenced by: pwsdiagmhm 18776 pwsco1mhm 18777 pwsco2mhm 18778 frlm0 21681 plypf1 26139 evls1fpws 33240 evlsvvval 41790 pwssplit4 42507 pwslnmlem2 42511 |
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