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Mirrors > Home > ILE Home > Th. List > eroprf2 | GIF version |
Description: Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) |
Ref | Expression |
---|---|
eropr2.1 | âĒ ð― = (ðī / âž ) |
eropr2.2 | âĒ âĻĢ = {âĻâĻðĨ, ðĶâĐ, ð§âĐ âĢ âð â ðī âð â ðī ((ðĨ = [ð] ➠⧠ðĶ = [ð] âž ) ⧠ð§ = [(ð + ð)] âž )} |
eropr2.3 | âĒ (ð â âž â ð) |
eropr2.4 | âĒ (ð â âž Er ð) |
eropr2.5 | âĒ (ð â ðī â ð) |
eropr2.6 | âĒ (ð â + :(ðī Ã ðī)âķðī) |
eropr2.7 | âĒ ((ð ⧠((ð â ðī ⧠ð â ðī) ⧠(ðĄ â ðī ⧠ðĒ â ðī))) â ((ð âž ð ⧠ðĄ âž ðĒ) â (ð + ðĄ) âž (ð + ðĒ))) |
Ref | Expression |
---|---|
eroprf2 | âĒ (ð â âĻĢ :(ð― Ã ð―)âķð―) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eropr2.1 | . 2 âĒ ð― = (ðī / âž ) | |
2 | eropr2.3 | . 2 âĒ (ð â âž â ð) | |
3 | eropr2.4 | . 2 âĒ (ð â âž Er ð) | |
4 | eropr2.5 | . 2 âĒ (ð â ðī â ð) | |
5 | eropr2.6 | . 2 âĒ (ð â + :(ðī Ã ðī)âķðī) | |
6 | eropr2.7 | . 2 âĒ ((ð ⧠((ð â ðī ⧠ð â ðī) ⧠(ðĄ â ðī ⧠ðĒ â ðī))) â ((ð âž ð ⧠ðĄ âž ðĒ) â (ð + ðĄ) âž (ð + ðĒ))) | |
7 | eropr2.2 | . 2 âĒ âĻĢ = {âĻâĻðĨ, ðĶâĐ, ð§âĐ âĢ âð â ðī âð â ðī ((ðĨ = [ð] ➠⧠ðĶ = [ð] âž ) ⧠ð§ = [(ð + ð)] âž )} | |
8 | 1, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 2, 2, 1 | eroprf 6618 | 1 âĒ (ð â âĻĢ :(ð― Ã ð―)âķð―) |
Colors of variables: wff set class |
Syntax hints: â wi 4 ⧠wa 104 = wceq 1353 â wcel 2146 âwrex 2454 â wss 3127 class class class wbr 3998 à cxp 4618 âķwf 5204 (class class class)co 5865 {coprab 5866 Er wer 6522 [cec 6523 / cqs 6524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-er 6525 df-ec 6527 df-qs 6531 |
This theorem is referenced by: (None) |
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