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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 3418 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
6 | prdsmndd.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
7 | 6 | elexd 3418 | . . . 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 18059 | . . 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 18060 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
13 | 2, 3 | mndid 18037 | . . . . 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 17991 | . . 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 235 | . 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 399 = wceq 1542 ∈ wcel 2114 ∀wral 3053 ∃wrex 3054 Vcvv 3398 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 +gcplusg 16668 0gc0g 16816 Xscprds 16822 Mndcmnd 18027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-hom 16692 df-cco 16693 df-0g 16818 df-prds 16824 df-mgm 17968 df-sgrp 18017 df-mnd 18028 |
This theorem is referenced by: pws0g 18063 prdspjmhm 18109 prdsgrpd 18327 prdsinvgd 18328 prds1 19486 dsmm0cl 20556 |
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