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Mirrors > Home > MPE Home > Th. List > eroprf2 | Structured version Visualization version 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] 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 |
---|---|
eroprf2 | âĒ (ð â âĻĢ :(ð― à 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 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, 1 | eroprf 8838 | 1 âĒ (ð â âĻĢ :(ð― à ð―)âķð―) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 394 = wceq 1533 â wcel 2098 âwrex 3066 â wss 3947 class class class wbr 5150 Ã cxp 5678 âķwf 6547 (class class class)co 7424 {coprab 7425 Er wer 8726 [cec 8727 / cqs 8728 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 7997 df-2nd 7998 df-er 8729 df-ec 8731 df-qs 8735 |
This theorem is referenced by: (None) |
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