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Theorem eroprf 8855
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 𝐽 = (𝐴 / 𝑅)
eropr.2 𝐾 = (𝐵 / 𝑆)
eropr.3 (𝜑𝑇𝑍)
eropr.4 (𝜑𝑅 Er 𝑈)
eropr.5 (𝜑𝑆 Er 𝑉)
eropr.6 (𝜑𝑇 Er 𝑊)
eropr.7 (𝜑𝐴𝑈)
eropr.8 (𝜑𝐵𝑉)
eropr.9 (𝜑𝐶𝑊)
eropr.10 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
eropr.11 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
eropr.12 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
eropr.13 (𝜑𝑅𝑋)
eropr.14 (𝜑𝑆𝑌)
eropr.15 𝐿 = (𝐶 / 𝑇)
Assertion
Ref Expression
eroprf (𝜑 :(𝐽 × 𝐾)⟶𝐿)
Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧,𝐴   𝐵,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐿,𝑝,𝑞,𝑥,𝑦,𝑧   𝐽,𝑝,𝑞,𝑥,𝑦,𝑧   𝑅,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝐾,𝑝,𝑞,𝑥,𝑦,𝑧   𝑆,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   + ,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝜑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑇,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑥,𝑦,𝑧   𝑋,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧   𝑌,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑧
Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   (𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑈(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝐽(𝑢,𝑡,𝑠,𝑟)   𝐾(𝑢,𝑡,𝑠,𝑟)   𝐿(𝑢,𝑡,𝑠,𝑟)   𝑉(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑊(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)   𝑍(𝑥,𝑦,𝑧,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝)

Proof of Theorem eroprf
StepHypRef Expression
1 eropr.3 . . . . . . . . . . . 12 (𝜑𝑇𝑍)
21ad2antrr 726 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → 𝑇𝑍)
3 eropr.10 . . . . . . . . . . . . 13 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
43adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → + :(𝐴 × 𝐵)⟶𝐶)
54fovcdmda 7604 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (𝑝 + 𝑞) ∈ 𝐶)
6 ecelqsg 8812 . . . . . . . . . . 11 ((𝑇𝑍 ∧ (𝑝 + 𝑞) ∈ 𝐶) → [(𝑝 + 𝑞)]𝑇 ∈ (𝐶 / 𝑇))
72, 5, 6syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → [(𝑝 + 𝑞)]𝑇 ∈ (𝐶 / 𝑇))
8 eropr.15 . . . . . . . . . 10 𝐿 = (𝐶 / 𝑇)
97, 8eleqtrrdi 2852 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → [(𝑝 + 𝑞)]𝑇𝐿)
10 eleq1a 2836 . . . . . . . . 9 ([(𝑝 + 𝑞)]𝑇𝐿 → (𝑧 = [(𝑝 + 𝑞)]𝑇𝑧𝐿))
119, 10syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (𝑧 = [(𝑝 + 𝑞)]𝑇𝑧𝐿))
1211adantld 490 . . . . . . 7 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧𝐿))
1312rexlimdvva 3213 . . . . . 6 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧𝐿))
1413abssdv 4068 . . . . 5 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → {𝑧 ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)} ⊆ 𝐿)
15 eropr.1 . . . . . . 7 𝐽 = (𝐴 / 𝑅)
16 eropr.2 . . . . . . 7 𝐾 = (𝐵 / 𝑆)
17 eropr.4 . . . . . . 7 (𝜑𝑅 Er 𝑈)
18 eropr.5 . . . . . . 7 (𝜑𝑆 Er 𝑉)
19 eropr.6 . . . . . . 7 (𝜑𝑇 Er 𝑊)
20 eropr.7 . . . . . . 7 (𝜑𝐴𝑈)
21 eropr.8 . . . . . . 7 (𝜑𝐵𝑉)
22 eropr.9 . . . . . . 7 (𝜑𝐶𝑊)
23 eropr.11 . . . . . . 7 ((𝜑 ∧ ((𝑟𝐴𝑠𝐴) ∧ (𝑡𝐵𝑢𝐵))) → ((𝑟𝑅𝑠𝑡𝑆𝑢) → (𝑟 + 𝑡)𝑇(𝑠 + 𝑢)))
2415, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23eroveu 8852 . . . . . 6 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ∃!𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
25 iotacl 6547 . . . . . 6 (∃!𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ {𝑧 ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)})
2624, 25syl 17 . . . . 5 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ {𝑧 ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)})
2714, 26sseldd 3984 . . . 4 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
2827ralrimivva 3202 . . 3 (𝜑 → ∀𝑥𝐽𝑦𝐾 (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
29 eqid 2737 . . . 4 (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))) = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)))
3029fmpo 8093 . . 3 (∀𝑥𝐽𝑦𝐾 (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿 ↔ (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿)
3128, 30sylib 218 . 2 (𝜑 → (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿)
32 eropr.12 . . . 4 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)}
3315, 16, 1, 17, 18, 19, 20, 21, 22, 3, 23, 32erovlem 8853 . . 3 (𝜑 = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
3433feq1d 6720 . 2 (𝜑 → ( :(𝐽 × 𝐾)⟶𝐿 ↔ (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿))
3531, 34mpbird 257 1 (𝜑 :(𝐽 × 𝐾)⟶𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ∃!weu 2568  {cab 2714  wral 3061  wrex 3070  wss 3951   class class class wbr 5143   × cxp 5683  cio 6512  wf 6557  (class class class)co 7431  {coprab 7432  cmpo 7433   Er wer 8742  [cec 8743   / cqs 8744
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-ec 8747  df-qs 8751
This theorem is referenced by:  eroprf2  8857
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