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Theorem xpsaddlem 16835
Description: Lemma for xpsadd 16836 and xpsmul 16837. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypotheses
Ref Expression
xpsval.t 𝑇 = (𝑅 ×s 𝑆)
xpsval.x 𝑋 = (Base‘𝑅)
xpsval.y 𝑌 = (Base‘𝑆)
xpsval.1 (𝜑𝑅𝑉)
xpsval.2 (𝜑𝑆𝑊)
xpsadd.3 (𝜑𝐴𝑋)
xpsadd.4 (𝜑𝐵𝑌)
xpsadd.5 (𝜑𝐶𝑋)
xpsadd.6 (𝜑𝐷𝑌)
xpsadd.7 (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)
xpsadd.8 (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
xpsaddlem.m · = (𝐸𝑅)
xpsaddlem.n × = (𝐸𝑆)
xpsaddlem.p = (𝐸𝑇)
xpsaddlem.f 𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
xpsaddlem.u 𝑈 = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
xpsaddlem.1 ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹) → ((𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) (𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
xpsaddlem.2 (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))
Assertion
Ref Expression
xpsaddlem (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Distinct variable groups:   𝑥,𝑘,𝑦,𝐴   𝐵,𝑘,𝑥,𝑦   𝐶,𝑘,𝑥,𝑦   𝐷,𝑘,𝑥,𝑦   𝑆,𝑘   𝑈,𝑘   𝑥,𝑊   𝜑,𝑘   · ,𝑘,𝑥,𝑦   × ,𝑘,𝑥,𝑦   𝑘,𝑋,𝑥,𝑦   𝑅,𝑘,𝑥   𝑘,𝑌,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑦)   𝑆(𝑥,𝑦)   (𝑥,𝑦,𝑘)   𝑇(𝑥,𝑦,𝑘)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦,𝑘)   𝐹(𝑥,𝑦,𝑘)   𝑉(𝑥,𝑦,𝑘)   𝑊(𝑦,𝑘)

Proof of Theorem xpsaddlem
StepHypRef Expression
1 df-ov 7141 . . . . 5 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2 xpsadd.3 . . . . . 6 (𝜑𝐴𝑋)
3 xpsadd.4 . . . . . 6 (𝜑𝐵𝑌)
4 xpsaddlem.f . . . . . . 7 𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
54xpsfval 16828 . . . . . 6 ((𝐴𝑋𝐵𝑌) → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
62, 3, 5syl2anc 587 . . . . 5 (𝜑 → (𝐴𝐹𝐵) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
71, 6syl5eqr 2873 . . . 4 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩})
82, 3opelxpd 5574 . . . . 5 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
94xpsff1o2 16831 . . . . . . 7 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
10 f1of 6596 . . . . . . 7 (𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹𝐹:(𝑋 × 𝑌)⟶ran 𝐹)
119, 10ax-mp 5 . . . . . 6 𝐹:(𝑋 × 𝑌)⟶ran 𝐹
1211ffvelrni 6831 . . . . 5 (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
138, 12syl 17 . . . 4 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ ran 𝐹)
147, 13eqeltrrd 2917 . . 3 (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹)
15 df-ov 7141 . . . . 5 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
16 xpsadd.5 . . . . . 6 (𝜑𝐶𝑋)
17 xpsadd.6 . . . . . 6 (𝜑𝐷𝑌)
184xpsfval 16828 . . . . . 6 ((𝐶𝑋𝐷𝑌) → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
1916, 17, 18syl2anc 587 . . . . 5 (𝜑 → (𝐶𝐹𝐷) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
2015, 19syl5eqr 2873 . . . 4 (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})
2116, 17opelxpd 5574 . . . . 5 (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
2211ffvelrni 6831 . . . . 5 (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
2321, 22syl 17 . . . 4 (𝜑 → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
2420, 23eqeltrrd 2917 . . 3 (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹)
25 xpsaddlem.1 . . 3 ((𝜑 ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ ran 𝐹 ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ ran 𝐹) → ((𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) (𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
2614, 24, 25mpd3an23 1460 . 2 (𝜑 → ((𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) (𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})))
27 f1ocnvfv 7017 . . . . 5 ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
289, 8, 27sylancr 590 . . . 4 (𝜑 → ((𝐹‘⟨𝐴, 𝐵⟩) = {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} → (𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩))
297, 28mpd 15 . . 3 (𝜑 → (𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) = ⟨𝐴, 𝐵⟩)
30 f1ocnvfv 7017 . . . . 5 ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
319, 21, 30sylancr 590 . . . 4 (𝜑 → ((𝐹‘⟨𝐶, 𝐷⟩) = {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} → (𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩))
3220, 31mpd 15 . . 3 (𝜑 → (𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = ⟨𝐶, 𝐷⟩)
3329, 32oveq12d 7156 . 2 (𝜑 → ((𝐹‘{⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}) (𝐹‘{⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (⟨𝐴, 𝐵𝐶, 𝐷⟩))
34 iftrue 4454 . . . . . . . . . . . 12 (𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑅)
3534fveq2d 6655 . . . . . . . . . . 11 (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸𝑅))
36 xpsaddlem.m . . . . . . . . . . 11 · = (𝐸𝑅)
3735, 36syl6eqr 2877 . . . . . . . . . 10 (𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = · )
38 iftrue 4454 . . . . . . . . . 10 (𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐴)
39 iftrue 4454 . . . . . . . . . 10 (𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐶)
4037, 38, 39oveq123d 7159 . . . . . . . . 9 (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐴 · 𝐶))
41 iftrue 4454 . . . . . . . . 9 (𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐴 · 𝐶))
4240, 41eqtr4d 2862 . . . . . . . 8 (𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
43 iffalse 4457 . . . . . . . . . . . 12 𝑘 = ∅ → if(𝑘 = ∅, 𝑅, 𝑆) = 𝑆)
4443fveq2d 6655 . . . . . . . . . . 11 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = (𝐸𝑆))
45 xpsaddlem.n . . . . . . . . . . 11 × = (𝐸𝑆)
4644, 45syl6eqr 2877 . . . . . . . . . 10 𝑘 = ∅ → (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)) = × )
47 iffalse 4457 . . . . . . . . . 10 𝑘 = ∅ → if(𝑘 = ∅, 𝐴, 𝐵) = 𝐵)
48 iffalse 4457 . . . . . . . . . 10 𝑘 = ∅ → if(𝑘 = ∅, 𝐶, 𝐷) = 𝐷)
4946, 47, 48oveq123d 7159 . . . . . . . . 9 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = (𝐵 × 𝐷))
50 iffalse 4457 . . . . . . . . 9 𝑘 = ∅ → if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)) = (𝐵 × 𝐷))
5149, 50eqtr4d 2862 . . . . . . . 8 𝑘 = ∅ → (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
5242, 51pm2.61i 185 . . . . . . 7 (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷))
53 xpsval.1 . . . . . . . . . . 11 (𝜑𝑅𝑉)
5453adantr 484 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2o) → 𝑅𝑉)
55 xpsval.2 . . . . . . . . . . 11 (𝜑𝑆𝑊)
5655adantr 484 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2o) → 𝑆𝑊)
57 simpr 488 . . . . . . . . . 10 ((𝜑𝑘 ∈ 2o) → 𝑘 ∈ 2o)
58 fvprif 16823 . . . . . . . . . 10 ((𝑅𝑉𝑆𝑊𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
5954, 56, 57, 58syl3anc 1368 . . . . . . . . 9 ((𝜑𝑘 ∈ 2o) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘) = if(𝑘 = ∅, 𝑅, 𝑆))
6059fveq2d 6655 . . . . . . . 8 ((𝜑𝑘 ∈ 2o) → (𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘)) = (𝐸‘if(𝑘 = ∅, 𝑅, 𝑆)))
612adantr 484 . . . . . . . . 9 ((𝜑𝑘 ∈ 2o) → 𝐴𝑋)
623adantr 484 . . . . . . . . 9 ((𝜑𝑘 ∈ 2o) → 𝐵𝑌)
63 fvprif 16823 . . . . . . . . 9 ((𝐴𝑋𝐵𝑌𝑘 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
6461, 62, 57, 63syl3anc 1368 . . . . . . . 8 ((𝜑𝑘 ∈ 2o) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘) = if(𝑘 = ∅, 𝐴, 𝐵))
6516adantr 484 . . . . . . . . 9 ((𝜑𝑘 ∈ 2o) → 𝐶𝑋)
6617adantr 484 . . . . . . . . 9 ((𝜑𝑘 ∈ 2o) → 𝐷𝑌)
67 fvprif 16823 . . . . . . . . 9 ((𝐶𝑋𝐷𝑌𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
6865, 66, 57, 67syl3anc 1368 . . . . . . . 8 ((𝜑𝑘 ∈ 2o) → ({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘) = if(𝑘 = ∅, 𝐶, 𝐷))
6960, 64, 68oveq123d 7159 . . . . . . 7 ((𝜑𝑘 ∈ 2o) → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) = (if(𝑘 = ∅, 𝐴, 𝐵)(𝐸‘if(𝑘 = ∅, 𝑅, 𝑆))if(𝑘 = ∅, 𝐶, 𝐷)))
70 xpsadd.7 . . . . . . . . 9 (𝜑 → (𝐴 · 𝐶) ∈ 𝑋)
7170adantr 484 . . . . . . . 8 ((𝜑𝑘 ∈ 2o) → (𝐴 · 𝐶) ∈ 𝑋)
72 xpsadd.8 . . . . . . . . 9 (𝜑 → (𝐵 × 𝐷) ∈ 𝑌)
7372adantr 484 . . . . . . . 8 ((𝜑𝑘 ∈ 2o) → (𝐵 × 𝐷) ∈ 𝑌)
74 fvprif 16823 . . . . . . . 8 (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
7571, 73, 57, 74syl3anc 1368 . . . . . . 7 ((𝜑𝑘 ∈ 2o) → ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘) = if(𝑘 = ∅, (𝐴 · 𝐶), (𝐵 × 𝐷)))
7652, 69, 753eqtr4a 2885 . . . . . 6 ((𝜑𝑘 ∈ 2o) → (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘)) = ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘))
7776mpteq2dva 5142 . . . . 5 (𝜑 → (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))) = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘)))
78 fnpr2o 16819 . . . . . . 7 ((𝑅𝑉𝑆𝑊) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
7953, 55, 78syl2anc 587 . . . . . 6 (𝜑 → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
80 xpsval.t . . . . . . . 8 𝑇 = (𝑅 ×s 𝑆)
81 xpsval.x . . . . . . . 8 𝑋 = (Base‘𝑅)
82 xpsval.y . . . . . . . 8 𝑌 = (Base‘𝑆)
83 eqid 2824 . . . . . . . 8 (Scalar‘𝑅) = (Scalar‘𝑅)
84 xpsaddlem.u . . . . . . . 8 𝑈 = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
8580, 81, 82, 53, 55, 4, 83, 84xpsrnbas 16833 . . . . . . 7 (𝜑 → ran 𝐹 = (Base‘𝑈))
8614, 85eleqtrd 2918 . . . . . 6 (𝜑 → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈))
8724, 85eleqtrd 2918 . . . . . 6 (𝜑 → {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈))
88 xpsaddlem.2 . . . . . 6 (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} ∈ (Base‘𝑈) ∧ {⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩} ∈ (Base‘𝑈)) → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))
8979, 86, 87, 88syl3anc 1368 . . . . 5 (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = (𝑘 ∈ 2o ↦ (({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩}‘𝑘)(𝐸‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘𝑘))({⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}‘𝑘))))
90 fnpr2o 16819 . . . . . . 7 (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
9170, 72, 90syl2anc 587 . . . . . 6 (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o)
92 dffn5 6705 . . . . . 6 ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} Fn 2o ↔ {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘)))
9391, 92sylib 221 . . . . 5 (𝜑 → {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} = (𝑘 ∈ 2o ↦ ({⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}‘𝑘)))
9477, 89, 933eqtr4d 2869 . . . 4 (𝜑 → ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩}) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
9594fveq2d 6655 . . 3 (𝜑 → (𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = (𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}))
96 df-ov 7141 . . . . 5 ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
974xpsfval 16828 . . . . . 6 (((𝐴 · 𝐶) ∈ 𝑋 ∧ (𝐵 × 𝐷) ∈ 𝑌) → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
9870, 72, 97syl2anc 587 . . . . 5 (𝜑 → ((𝐴 · 𝐶)𝐹(𝐵 × 𝐷)) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
9996, 98syl5eqr 2873 . . . 4 (𝜑 → (𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩})
10070, 72opelxpd 5574 . . . . 5 (𝜑 → ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌))
101 f1ocnvfv 7017 . . . . 5 ((𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 ∧ ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩ ∈ (𝑋 × 𝑌)) → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} → (𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
1029, 100, 101sylancr 590 . . . 4 (𝜑 → ((𝐹‘⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩) = {⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩} → (𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩))
10399, 102mpd 15 . . 3 (𝜑 → (𝐹‘{⟨∅, (𝐴 · 𝐶)⟩, ⟨1o, (𝐵 × 𝐷)⟩}) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
10495, 103eqtrd 2859 . 2 (𝜑 → (𝐹‘({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} (𝐸𝑈){⟨∅, 𝐶⟩, ⟨1o, 𝐷⟩})) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
10526, 33, 1043eqtr3d 2867 1 (𝜑 → (⟨𝐴, 𝐵𝐶, 𝐷⟩) = ⟨(𝐴 · 𝐶), (𝐵 × 𝐷)⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  c0 4274  ifcif 4448  {cpr 4550  cop 4554  cmpt 5127   × cxp 5534  ccnv 5535  ran crn 5537   Fn wfn 6331  wf 6332  1-1-ontowf1o 6335  cfv 6336  (class class class)co 7138  cmpo 7140  1oc1o 8078  2oc2o 8079  Basecbs 16472  Scalarcsca 16557  Xscprds 16708   ×s cxps 16768
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-2 11686  df-3 11687  df-4 11688  df-5 11689  df-6 11690  df-7 11691  df-8 11692  df-9 11693  df-n0 11884  df-z 11968  df-dec 12085  df-uz 12230  df-fz 12884  df-struct 16474  df-ndx 16475  df-slot 16476  df-base 16478  df-plusg 16567  df-mulr 16568  df-sca 16570  df-vsca 16571  df-ip 16572  df-tset 16573  df-ple 16574  df-ds 16576  df-hom 16578  df-cco 16579  df-prds 16710
This theorem is referenced by:  xpsadd  16836  xpsmul  16837
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