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Mirrors > Home > MPE Home > Th. List > cntri | Structured version Visualization version GIF version |
Description: Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
cntri.b | ⊢ 𝐵 = (Base‘𝑀) |
cntri.p | ⊢ + = (+g‘𝑀) |
cntri.z | ⊢ 𝑍 = (Cntr‘𝑀) |
Ref | Expression |
---|---|
cntri | ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cntri.z | . . . 4 ⊢ 𝑍 = (Cntr‘𝑀) | |
2 | cntri.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
3 | eqid 2823 | . . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀) | |
4 | 2, 3 | cntrval 18451 | . . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀) |
5 | 1, 4 | eqtr4i 2849 | . . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵) |
6 | 5 | eleq2i 2906 | . 2 ⊢ (𝑋 ∈ 𝑍 ↔ 𝑋 ∈ ((Cntz‘𝑀)‘𝐵)) |
7 | cntri.p | . . 3 ⊢ + = (+g‘𝑀) | |
8 | 7, 3 | cntzi 18461 | . 2 ⊢ ((𝑋 ∈ ((Cntz‘𝑀)‘𝐵) ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
9 | 6, 8 | sylanb 583 | 1 ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Cntzccntz 18447 Cntrccntr 18448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-cntz 18449 df-cntr 18450 |
This theorem is referenced by: cntrcmnd 18964 primefld 19586 |
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