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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 2727 | . . . . 5 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
3 | eqid 2727 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | 2, 3 | mgpbas 20085 | . . . 4 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾)) |
5 | 1, 4 | eqtrdi 2783 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾))) |
6 | rngidpropd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
7 | eqid 2727 | . . . . 5 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
8 | eqid 2727 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
9 | 7, 8 | mgpbas 20085 | . . . 4 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿)) |
10 | 6, 9 | eqtrdi 2783 | . . 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 2727 | . . . . . 6 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
13 | 2, 12 | mgpplusg 20083 | . . . . 5 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
14 | 13 | oveqi 7437 | . . . 4 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦) |
15 | eqid 2727 | . . . . . 6 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
16 | 7, 15 | mgpplusg 20083 | . . . . 5 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
17 | 16 | oveqi 7437 | . . . 4 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦) |
18 | 11, 14, 17 | 3eqtr3g 2790 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
19 | 5, 10, 18 | grpidpropd 18627 | . 2 ⊢ (𝜑 → (0g‘(mulGrp‘𝐾)) = (0g‘(mulGrp‘𝐿))) |
20 | eqid 2727 | . . 3 ⊢ (1r‘𝐾) = (1r‘𝐾) | |
21 | 2, 20 | ringidval 20128 | . 2 ⊢ (1r‘𝐾) = (0g‘(mulGrp‘𝐾)) |
22 | eqid 2727 | . . 3 ⊢ (1r‘𝐿) = (1r‘𝐿) | |
23 | 7, 22 | ringidval 20128 | . 2 ⊢ (1r‘𝐿) = (0g‘(mulGrp‘𝐿)) |
24 | 19, 21, 23 | 3eqtr4g 2792 | 1 ⊢ (𝜑 → (1r‘𝐾) = (1r‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ‘cfv 6551 (class class class)co 7424 Basecbs 17185 +gcplusg 17238 .rcmulr 17239 0gc0g 17426 mulGrpcmgp 20079 1rcur 20126 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-plusg 17251 df-0g 17428 df-mgp 20080 df-ur 20127 |
This theorem is referenced by: unitpropd 20361 subrgpropd 20552 lmodprop2d 20812 opsr1 22143 ply1mpl1 22181 sra1r 33287 zlm1 33567 hlhils1N 41427 |
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