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Theorem eroprf 8635
Description: Functionality of an operation defined on equivalence classes. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
eropr.1 ð― = (ðī / 𝑅)
eropr.2 ðū = (ðĩ / 𝑆)
eropr.3 (𝜑 → 𝑇 ∈ 𝑍)
eropr.4 (𝜑 → 𝑅 Er 𝑈)
eropr.5 (𝜑 → 𝑆 Er 𝑉)
eropr.6 (𝜑 → 𝑇 Er 𝑊)
eropr.7 (𝜑 → ðī ⊆ 𝑈)
eropr.8 (𝜑 → ðĩ ⊆ 𝑉)
eropr.9 (𝜑 → ðķ ⊆ 𝑊)
eropr.10 (𝜑 → + :(ðī × ðĩ)âŸķðķ)
eropr.11 ((𝜑 ∧ ((𝑟 ∈ ðī ∧ 𝑠 ∈ ðī) ∧ (ð‘Ą ∈ ðĩ ∧ ð‘Ē ∈ ðĩ))) → ((𝑟𝑅𝑠 ∧ ð‘Ąð‘†ð‘Ē) → (𝑟 + ð‘Ą)𝑇(𝑠 + ð‘Ē)))
eropr.12 âĻĢ = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
eropr.13 (𝜑 → 𝑅 ∈ 𝑋)
eropr.14 (𝜑 → 𝑆 ∈ 𝑌)
eropr.15 ðŋ = (ðķ / 𝑇)
Assertion
Ref Expression
eroprf (𝜑 → âĻĢ :(ð― × ðū)âŸķðŋ)
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧,ðī   ðĩ,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   ðŋ,𝑝,𝑞,ð‘Ĩ,ð‘Ķ,𝑧   ð―,𝑝,𝑞,ð‘Ĩ,ð‘Ķ,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   ðū,𝑝,𝑞,ð‘Ĩ,ð‘Ķ,𝑧   𝑆,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   + ,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   𝑇,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,ð‘Ĩ,ð‘Ķ,𝑧   𝑋,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,𝑧   𝑌,𝑝,𝑞,𝑟,𝑠,ð‘Ą,ð‘Ē,𝑧
Allowed substitution hints:   ðķ(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   âĻĢ (ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   𝑈(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   ð―(ð‘Ē,ð‘Ą,𝑠,𝑟)   ðū(ð‘Ē,ð‘Ą,𝑠,𝑟)   ðŋ(ð‘Ē,ð‘Ą,𝑠,𝑟)   𝑉(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   𝑊(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)   𝑋(ð‘Ĩ,ð‘Ķ)   𝑌(ð‘Ĩ,ð‘Ķ)   𝑍(ð‘Ĩ,ð‘Ķ,𝑧,ð‘Ē,ð‘Ą,𝑠,𝑟,𝑞,𝑝)

Proof of Theorem eroprf
StepHypRef Expression
1 eropr.3 . . . . . . . . . . . 12 (𝜑 → 𝑇 ∈ 𝑍)
21ad2antrr 724 . . . . . . . . . . 11 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → 𝑇 ∈ 𝑍)
3 eropr.10 . . . . . . . . . . . . 13 (𝜑 → + :(ðī × ðĩ)âŸķðķ)
43adantr 482 . . . . . . . . . . . 12 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → + :(ðī × ðĩ)âŸķðķ)
54fovcdmda 7475 . . . . . . . . . . 11 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → (𝑝 + 𝑞) ∈ ðķ)
6 ecelqsg 8592 . . . . . . . . . . 11 ((𝑇 ∈ 𝑍 ∧ (𝑝 + 𝑞) ∈ ðķ) → [(𝑝 + 𝑞)]𝑇 ∈ (ðķ / 𝑇))
72, 5, 6syl2anc 585 . . . . . . . . . 10 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → [(𝑝 + 𝑞)]𝑇 ∈ (ðķ / 𝑇))
8 eropr.15 . . . . . . . . . 10 ðŋ = (ðķ / 𝑇)
97, 8eleqtrrdi 2848 . . . . . . . . 9 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → [(𝑝 + 𝑞)]𝑇 ∈ ðŋ)
10 eleq1a 2832 . . . . . . . . 9 ([(𝑝 + 𝑞)]𝑇 ∈ ðŋ → (𝑧 = [(𝑝 + 𝑞)]𝑇 → 𝑧 ∈ ðŋ))
119, 10syl 17 . . . . . . . 8 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → (𝑧 = [(𝑝 + 𝑞)]𝑇 → 𝑧 ∈ ðŋ))
1211adantld 492 . . . . . . 7 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → (((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧 ∈ ðŋ))
1312rexlimdvva 3202 . . . . . 6 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → (∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧 ∈ ðŋ))
1413abssdv 4007 . . . . 5 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → {𝑧 âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} ⊆ ðŋ)
15 eropr.1 . . . . . . 7 ð― = (ðī / 𝑅)
16 eropr.2 . . . . . . 7 ðū = (ðĩ / 𝑆)
17 eropr.4 . . . . . . 7 (𝜑 → 𝑅 Er 𝑈)
18 eropr.5 . . . . . . 7 (𝜑 → 𝑆 Er 𝑉)
19 eropr.6 . . . . . . 7 (𝜑 → 𝑇 Er 𝑊)
20 eropr.7 . . . . . . 7 (𝜑 → ðī ⊆ 𝑈)
21 eropr.8 . . . . . . 7 (𝜑 → ðĩ ⊆ 𝑉)
22 eropr.9 . . . . . . 7 (𝜑 → ðķ ⊆ 𝑊)
23 eropr.11 . . . . . . 7 ((𝜑 ∧ ((𝑟 ∈ ðī ∧ 𝑠 ∈ ðī) ∧ (ð‘Ą ∈ ðĩ ∧ ð‘Ē ∈ ðĩ))) → ((𝑟𝑅𝑠 ∧ ð‘Ąð‘†ð‘Ē) → (𝑟 + ð‘Ą)𝑇(𝑠 + ð‘Ē)))
2415, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23eroveu 8632 . . . . . 6 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → ∃!𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
25 iotacl 6444 . . . . . 6 (∃!𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ {𝑧 âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)})
2624, 25syl 17 . . . . 5 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ {𝑧 âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)})
2714, 26sseldd 3927 . . . 4 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ ðŋ)
2827ralrimivva 3194 . . 3 (𝜑 → ∀ð‘Ĩ ∈ ð― ∀ð‘Ķ ∈ ðū (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ ðŋ)
29 eqid 2736 . . . 4 (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) = (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
3029fmpo 7940 . . 3 (∀ð‘Ĩ ∈ ð― ∀ð‘Ķ ∈ ðū (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ ðŋ ↔ (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(ð― × ðū)âŸķðŋ)
3128, 30sylib 217 . 2 (𝜑 → (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(ð― × ðū)âŸķðŋ)
32 eropr.12 . . . 4 âĻĢ = {âŸĻâŸĻð‘Ĩ, ð‘ĶâŸĐ, 𝑧âŸĐ âˆĢ ∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
3315, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23, 32erovlem 8633 . . 3 (𝜑 → âĻĢ = (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
3433feq1d 6615 . 2 (𝜑 → ( âĻĢ :(ð― × ðū)âŸķðŋ ↔ (ð‘Ĩ ∈ ð―, ð‘Ķ ∈ ðū â†Ķ (â„Đ𝑧∃𝑝 ∈ ðī ∃𝑞 ∈ ðĩ ((ð‘Ĩ = [𝑝]𝑅 ∧ ð‘Ķ = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(ð― × ðū)âŸķðŋ))
3531, 34mpbird 257 1 (𝜑 → âĻĢ :(ð― × ðū)âŸķðŋ)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  âˆƒ!weu 2566  {cab 2713  âˆ€wral 3062  âˆƒwrex 3071   ⊆ wss 3892   class class class wbr 5081   × cxp 5598  â„Đcio 6408  âŸķwf 6454  (class class class)co 7307  {coprab 7308   ∈ cmpo 7309   Er wer 8526  [cec 8527   / cqs 8528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864  df-er 8529  df-ec 8531  df-qs 8535
This theorem is referenced by:  eroprf2  8637
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