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Mirrors > Home > MPE Home > Th. List > erov2 | Structured version Visualization version GIF version |
Description: The value 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 |
---|---|
erov2 | âĒ ((ð ⧠ð â ðī ⧠ð â ðī) â ([ð] âž âĻĢ [ð] âž ) = [(ð + ð)] âž ) |
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 | erov 8754 | 1 âĒ ((ð ⧠ð â ðī ⧠ð â ðī) â ([ð] âž âĻĢ [ð] âž ) = [(ð + ð)] âž ) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 ⧠wa 397 ⧠w3a 1088 = wceq 1542 â wcel 2107 âwrex 3074 â wss 3911 class class class wbr 5106 à cxp 5632 âķwf 6493 (class class class)co 7358 {coprab 7359 Er wer 8646 [cec 8647 / cqs 8648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8649 df-ec 8651 df-qs 8655 |
This theorem is referenced by: (None) |
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