Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlm1 | Structured version Visualization version GIF version |
Description: Unit of a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
zlmlem2.1 | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlm1.1 | ⊢ 1 = (1r‘𝐺) |
Ref | Expression |
---|---|
zlm1 | ⊢ 1 = (1r‘𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlm1.1 | . 2 ⊢ 1 = (1r‘𝐺) | |
2 | eqid 2738 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝐺) = (Base‘𝐺)) |
4 | zlmlem2.1 | . . . . . 6 ⊢ 𝑊 = (ℤMod‘𝐺) | |
5 | 4, 2 | zlmbas 20340 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝑊) |
6 | 5 | a1i 11 | . . . 4 ⊢ (⊤ → (Base‘𝐺) = (Base‘𝑊)) |
7 | eqid 2738 | . . . . . . 7 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
8 | 4, 7 | zlmmulr 20342 | . . . . . 6 ⊢ (.r‘𝐺) = (.r‘𝑊) |
9 | 8 | a1i 11 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (.r‘𝐺) = (.r‘𝑊)) |
10 | 9 | oveqd 7189 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(.r‘𝐺)𝑦) = (𝑥(.r‘𝑊)𝑦)) |
11 | 3, 6, 10 | rngidpropd 19569 | . . 3 ⊢ (⊤ → (1r‘𝐺) = (1r‘𝑊)) |
12 | 11 | mptru 1549 | . 2 ⊢ (1r‘𝐺) = (1r‘𝑊) |
13 | 1, 12 | eqtri 2761 | 1 ⊢ 1 = (1r‘𝑊) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ‘cfv 6339 Basecbs 16588 .rcmulr 16671 1rcur 19372 ℤModczlm 20323 |
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-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 |
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-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 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-om 7602 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-er 8322 df-en 8558 df-dom 8559 df-sdom 8560 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-plusg 16683 df-mulr 16684 df-sca 16686 df-vsca 16687 df-0g 16820 df-mgp 19361 df-ur 19373 df-zlm 20327 |
This theorem is referenced by: zrhnm 31491 |
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