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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 8826 | 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 1085 = wceq 1534 â wcel 2099 âwrex 3066 â wss 3945 class class class wbr 5142 Ã cxp 5670 âķwf 6538 (class class class)co 7414 {coprab 7415 Er wer 8715 [cec 8716 / cqs 8717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8718 df-ec 8720 df-qs 8724 |
This theorem is referenced by: (None) |
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