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Theorem eroprf 8754
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 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → 𝑇𝑍)
3 eropr.10 . . . . . . . . . . . . 13 (𝜑+ :(𝐴 × 𝐵)⟶𝐶)
43adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → + :(𝐴 × 𝐵)⟶𝐶)
54fovcdmda 7525 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (𝑝 + 𝑞) ∈ 𝐶)
6 ecelqsg 8711 . . . . . . . . . . 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 2849 . . . . . . . . 9 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → [(𝑝 + 𝑞)]𝑇𝐿)
10 eleq1a 2833 . . . . . . . . 9 ([(𝑝 + 𝑞)]𝑇𝐿 → (𝑧 = [(𝑝 + 𝑞)]𝑇𝑧𝐿))
119, 10syl 17 . . . . . . . 8 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (𝑧 = [(𝑝 + 𝑞)]𝑇𝑧𝐿))
1211adantld 491 . . . . . . 7 (((𝜑 ∧ (𝑥𝐽𝑦𝐾)) ∧ (𝑝𝐴𝑞𝐵)) → (((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧𝐿))
1312rexlimdvva 3205 . . . . . 6 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (∃𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇) → 𝑧𝐿))
1413abssdv 4025 . . . . 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 8751 . . . . . 6 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → ∃!𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))
25 iotacl 6482 . . . . . 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 3945 . . . 4 ((𝜑 ∧ (𝑥𝐽𝑦𝐾)) → (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
2827ralrimivva 3197 . . 3 (𝜑 → ∀𝑥𝐽𝑦𝐾 (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇)) ∈ 𝐿)
29 eqid 2736 . . . 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 8000 . . 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 217 . 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 8752 . . 3 (𝜑 = (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))))
3433feq1d 6653 . 2 (𝜑 → ( :(𝐽 × 𝐾)⟶𝐿 ↔ (𝑥𝐽, 𝑦𝐾 ↦ (℩𝑧𝑝𝐴𝑞𝐵 ((𝑥 = [𝑝]𝑅𝑦 = [𝑞]𝑆) ∧ 𝑧 = [(𝑝 + 𝑞)]𝑇))):(𝐽 × 𝐾)⟶𝐿))
3531, 34mpbird 256 1 (𝜑 :(𝐽 × 𝐾)⟶𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ∃!weu 2566  {cab 2713  wral 3064  wrex 3073  wss 3910   class class class wbr 5105   × cxp 5631  cio 6446  wf 6492  (class class class)co 7357  {coprab 7358  cmpo 7359   Er wer 8645  [cec 8646   / cqs 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-ec 8650  df-qs 8654
This theorem is referenced by:  eroprf2  8756
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