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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] âž â§ ðĶ = [ð] âž ) â§ 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 à ðī)âķðī) |
eropr2.7 | âĒ ((ð â§ ((ð â ðī â§ ð â ðī) â§ (ðĄ â ðī â§ ðĒ â ðī))) â ((ð âž ð â§ ðĄ âž ðĒ) â (ð + ðĄ) âž (ð + ðĒ))) |
Ref | Expression |
---|---|
erov2 | âĒ ((ð â§ ð â ðī â§ 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 à ðī)âķðī) | |
6 | eropr2.7 | . 2 âĒ ((ð â§ ((ð â ðī â§ ð â ðī) â§ (ðĄ â ðī â§ ðĒ â ðī))) â ((ð âž ð â§ 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] âž â§ ðĶ = [ð] âž ) â§ ð§ = [(ð + ð)] âž )} | |
8 | 1, 1, 2, 3, 3, 3, 4, 4, 4, 5, 6, 7, 2, 2 | erov 8805 | 1 âĒ ((ð â§ ð â ðī â§ ð â ðī) â ([ð] âž âĻĢ [ð] âž ) = [(ð + ð)] âž ) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 395 â§ w3a 1084 = wceq 1533 â wcel 2098 âwrex 3062 â wss 3941 class class class wbr 5139 Ã cxp 5665 âķwf 6530 (class class class)co 7402 {coprab 7403 Er wer 8697 [cec 8698 / cqs 8699 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-ec 8702 df-qs 8706 |
This theorem is referenced by: (None) |
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