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Mirrors > Home > MPE Home > Th. List > rngidpropd | Structured version Visualization version GIF version |
Description: The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) |
Ref | Expression |
---|---|
rngidpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
rngidpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
rngidpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
rngidpropd | ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rngidpropd.1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | eqid 2798 | . . . . 5 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
3 | eqid 2798 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | 2, 3 | mgpbas 19238 | . . . 4 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾)) |
5 | 1, 4 | eqtrdi 2849 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾))) |
6 | rngidpropd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
7 | eqid 2798 | . . . . 5 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
8 | eqid 2798 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
9 | 7, 8 | mgpbas 19238 | . . . 4 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿)) |
10 | 6, 9 | eqtrdi 2849 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿))) |
11 | rngidpropd.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
12 | eqid 2798 | . . . . . 6 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
13 | 2, 12 | mgpplusg 19236 | . . . . 5 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
14 | 13 | oveqi 7148 | . . . 4 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦) |
15 | eqid 2798 | . . . . . 6 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
16 | 7, 15 | mgpplusg 19236 | . . . . 5 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
17 | 16 | oveqi 7148 | . . . 4 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦) |
18 | 11, 14, 17 | 3eqtr3g 2856 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
19 | 5, 10, 18 | grpidpropd 17864 | . 2 ⊢ (𝜑 → (0g‘(mulGrp‘𝐾)) = (0g‘(mulGrp‘𝐿))) |
20 | eqid 2798 | . . 3 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
21 | 2, 20 | ringidval 19246 | . 2 ⊢ (1r‘𝐾) = (0g‘(mulGrp‘𝐾)) |
22 | eqid 2798 | . . 3 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
23 | 7, 22 | ringidval 19246 | . 2 ⊢ (1r‘𝐿) = (0g‘(mulGrp‘𝐿)) |
24 | 19, 21, 23 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 mulGrpcmgp 19232 1rcur 19244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-0g 16707 df-mgp 19233 df-ur 19245 |
This theorem is referenced by: unitpropd 19443 subrgpropd 19563 lmodprop2d 19689 opsr1 20848 ply1mpl1 20886 sra1r 31074 zlm1 31314 hlhils1N 39242 |
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