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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 2738 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
3 | eqid 2738 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
4 | prdsmndd.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | 4 | elexd 3452 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
6 | prdsmndd.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 6 | elexd 3452 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
8 | prdsmndd.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) | |
9 | eqid 2738 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
10 | 1, 2, 3, 5, 7, 8, 9 | prdsidlem 18417 | . . 3 ⊢ (𝜑 → ((0g ∘ 𝑅) ∈ (Base‘𝑌) ∧ ∀𝑏 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)(0g ∘ 𝑅)) = 𝑏))) |
11 | eqid 2738 | . . . 4 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
12 | 1, 6, 4, 8 | prdsmndd 18418 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | 2, 3 | mndid 18395 | . . . . 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 18349 | . . 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 231 | . 2 ⊢ (𝜑 → (0g‘𝑌) = (0g ∘ 𝑅)) |
17 | 16 | eqcomd 2744 | 1 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 0gc0g 17150 Xscprds 17156 Mndcmnd 18385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-prds 17158 df-mgm 18326 df-sgrp 18375 df-mnd 18386 |
This theorem is referenced by: pws0g 18421 prdspjmhm 18467 prdsgrpd 18685 prdsinvgd 18686 prds1 19853 dsmm0cl 20947 |
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