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Mirrors > Home > MPE Home > Th. List > rngidpropd | Structured version Visualization version GIF version |
Description: The ring unity 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 2725 | . . . . 5 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
3 | eqid 2725 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | 2, 3 | mgpbas 20084 | . . . 4 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾)) |
5 | 1, 4 | eqtrdi 2781 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾))) |
6 | rngidpropd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
7 | eqid 2725 | . . . . 5 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
8 | eqid 2725 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
9 | 7, 8 | mgpbas 20084 | . . . 4 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿)) |
10 | 6, 9 | eqtrdi 2781 | . . 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 2725 | . . . . . 6 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
13 | 2, 12 | mgpplusg 20082 | . . . . 5 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
14 | 13 | oveqi 7429 | . . . 4 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦) |
15 | eqid 2725 | . . . . . 6 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
16 | 7, 15 | mgpplusg 20082 | . . . . 5 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
17 | 16 | oveqi 7429 | . . . 4 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦) |
18 | 11, 14, 17 | 3eqtr3g 2788 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
19 | 5, 10, 18 | grpidpropd 18621 | . 2 ⊢ (𝜑 → (0g‘(mulGrp‘𝐾)) = (0g‘(mulGrp‘𝐿))) |
20 | eqid 2725 | . . 3 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
21 | 2, 20 | ringidval 20127 | . 2 ⊢ (1r‘𝐾) = (0g‘(mulGrp‘𝐾)) |
22 | eqid 2725 | . . 3 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
23 | 7, 22 | ringidval 20127 | . 2 ⊢ (1r‘𝐿) = (0g‘(mulGrp‘𝐿)) |
24 | 19, 21, 23 | 3eqtr4g 2790 | 1 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 +gcplusg 17232 .rcmulr 17233 0gc0g 17420 mulGrpcmgp 20078 1rcur 20125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgp 20079 df-ur 20126 |
This theorem is referenced by: unitpropd 20360 subrgpropd 20551 lmodprop2d 20811 opsr1 22147 ply1mpl1 22185 sra1r 33339 zlm1 33619 hlhils1N 41479 |
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