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Mirrors > Home > MPE Home > Th. List > prds0g | Structured version Visualization version GIF version |
Description: Zero in a product of monoids. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
Ref | Expression |
---|---|
prdsmndd.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsmndd.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsmndd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsmndd.r | ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) |
Ref | Expression |
---|---|
prds0g | ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsmndd.y | . . . 4 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | eqid 2778 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
3 | eqid 2778 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
4 | prdsmndd.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | 4 | elexd 3416 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
6 | prdsmndd.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 6 | elexd 3416 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
8 | prdsmndd.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) | |
9 | eqid 2778 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
10 | 1, 2, 3, 5, 7, 8, 9 | prdsidlem 17708 | . . 3 ⊢ (𝜑 → ((0g ∘ 𝑅) ∈ (Base‘𝑌) ∧ ∀𝑏 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)(0g ∘ 𝑅)) = 𝑏))) |
11 | eqid 2778 | . . . 4 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
12 | 1, 6, 4, 8 | prdsmndd 17709 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | 2, 3 | mndid 17689 | . . . . 5 ⊢ (𝑌 ∈ Mnd → ∃𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝑎(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)𝑎) = 𝑏)) |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → ∃𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝑎(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)𝑎) = 𝑏)) |
15 | 2, 11, 3, 14 | ismgmid 17650 | . . 3 ⊢ (𝜑 → (((0g ∘ 𝑅) ∈ (Base‘𝑌) ∧ ∀𝑏 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)(0g ∘ 𝑅)) = 𝑏)) ↔ (0g‘𝑌) = (0g ∘ 𝑅))) |
16 | 10, 15 | mpbid 224 | . 2 ⊢ (𝜑 → (0g‘𝑌) = (0g ∘ 𝑅)) |
17 | 16 | eqcomd 2784 | 1 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ∘ ccom 5359 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 0gc0g 16486 Xscprds 16492 Mndcmnd 17680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-hom 16362 df-cco 16363 df-0g 16488 df-prds 16494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 |
This theorem is referenced by: pws0g 17712 prdspjmhm 17753 prdsgrpd 17912 prdsinvgd 17913 prds1 19001 dsmm0cl 20483 |
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