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Theorem eroprf 6628
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 488 . . . . . . . . . . 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 276 . . . . . . . . . . . 12 ((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) → + :(ðī × ðĩ)âŸķðķ)
54fovcdmda 6018 . . . . . . . . . . 11 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → (𝑝 + 𝑞) ∈ ðķ)
6 ecelqsg 6588 . . . . . . . . . . 11 ((𝑇 ∈ 𝑍 ∧ (𝑝 + 𝑞) ∈ ðķ) → [(𝑝 + 𝑞)]𝑇 ∈ (ðķ / 𝑇))
72, 5, 6syl2anc 411 . . . . . . . . . 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 2271 . . . . . . . . 9 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → [(𝑝 + 𝑞)]𝑇 ∈ ðŋ)
10 eleq1a 2249 . . . . . . . . 9 ([(𝑝 + 𝑞)]𝑇 ∈ ðŋ → (𝑧 = [(𝑝 + 𝑞)]𝑇 → 𝑧 ∈ ðŋ))
119, 10syl 14 . . . . . . . 8 (((𝜑 ∧ (ð‘Ĩ ∈ ð― ∧ ð‘Ķ ∈ ðū)) ∧ (𝑝 ∈ ðī ∧ 𝑞 ∈ ðĩ)) → (𝑧 = [(𝑝 + 𝑞)]𝑇 → 𝑧 ∈ ðŋ))
1211adantld 278 . . . . . . 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 2602 . . . . . 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 3230 . . . . 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 6626 . . . . . 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 5202 . . . . . 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 14 . . . . 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 3157 . . . 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 2559 . . 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 2177 . . . 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 6202 . . 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 122 . 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 6627 . . 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 5353 . 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 167 1 (𝜑 → âĻĢ :(ð― × ðū)âŸķðŋ)
Colors of variables: wff set class
Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353  âˆƒ!weu 2026   ∈ wcel 2148  {cab 2163  âˆ€wral 2455  âˆƒwrex 2456   ⊆ wss 3130   class class class wbr 4004   × cxp 4625  â„Đcio 5177  âŸķwf 5213  (class class class)co 5875  {coprab 5876   ∈ cmpo 5877   Er wer 6532  [cec 6533   / cqs 6534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-er 6535  df-ec 6537  df-qs 6541
This theorem is referenced by:  eroprf2  6629
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