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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 2818 | . . . . 5 ⊢ (Cntz‘𝑀) = (Cntz‘𝑀) | |
4 | 2, 3 | cntrval 18387 | . . . 4 ⊢ ((Cntz‘𝑀)‘𝐵) = (Cntr‘𝑀) |
5 | 1, 4 | eqtr4i 2844 | . . 3 ⊢ 𝑍 = ((Cntz‘𝑀)‘𝐵) |
6 | 5 | eleq2i 2901 | . 2 ⊢ (𝑋 ∈ 𝑍 ↔ 𝑋 ∈ ((Cntz‘𝑀)‘𝐵)) |
7 | cntri.p | . . 3 ⊢ + = (+g‘𝑀) | |
8 | 7, 3 | cntzi 18397 | . 2 ⊢ ((𝑋 ∈ ((Cntz‘𝑀)‘𝐵) ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
9 | 6, 8 | sylanb 581 | 1 ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Cntzccntz 18383 Cntrccntr 18384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-cntz 18385 df-cntr 18386 |
This theorem is referenced by: cntrcmnd 18891 primefld 19513 |
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