Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2610 | . . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌) | |
3 | eqid 2610 | . . . 4 ⊢ (+g‘𝑌) = (+g‘𝑌) | |
4 | prdsmndd.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | elex 3185 | . . . . 5 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ V) |
7 | prdsmndd.i | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
8 | elex 3185 | . . . . 5 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ V) |
10 | prdsmndd.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) | |
11 | eqid 2610 | . . . 4 ⊢ (0g ∘ 𝑅) = (0g ∘ 𝑅) | |
12 | 1, 2, 3, 6, 9, 10, 11 | prdsidlem 17145 | . . 3 ⊢ (𝜑 → ((0g ∘ 𝑅) ∈ (Base‘𝑌) ∧ ∀𝑏 ∈ (Base‘𝑌)(((0g ∘ 𝑅)(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)(0g ∘ 𝑅)) = 𝑏))) |
13 | eqid 2610 | . . . 4 ⊢ (0g‘𝑌) = (0g‘𝑌) | |
14 | 1, 7, 4, 10 | prdsmndd 17146 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Mnd) |
15 | 2, 3 | mndid 17126 | . . . . 5 ⊢ (𝑌 ∈ Mnd → ∃𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝑎(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)𝑎) = 𝑏)) |
16 | 14, 15 | syl 17 | . . . 4 ⊢ (𝜑 → ∃𝑎 ∈ (Base‘𝑌)∀𝑏 ∈ (Base‘𝑌)((𝑎(+g‘𝑌)𝑏) = 𝑏 ∧ (𝑏(+g‘𝑌)𝑎) = 𝑏)) |
17 | 2, 13, 3, 16 | ismgmid 17087 | . . 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))) |
18 | 12, 17 | mpbid 221 | . 2 ⊢ (𝜑 → (0g‘𝑌) = (0g ∘ 𝑅)) |
19 | 18 | eqcomd 2616 | 1 ⊢ (𝜑 → (0g ∘ 𝑅) = (0g‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Xscprds 15929 Mndcmnd 17117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-0g 15925 df-prds 15931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 |
This theorem is referenced by: pws0g 17149 prdspjmhm 17190 prdsgrpd 17348 prdsinvgd 17349 prds1 18437 dsmm0cl 19903 |
Copyright terms: Public domain | W3C validator |