Theorem xpsval 13557

Description: Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)

Hypotheses

Ref	Expression
xpsval.t	⊢ 𝑇 = (𝑅 ×s 𝑆)
xpsval.x	⊢ 𝑋 = (Base‘𝑅)
xpsval.y	⊢ 𝑌 = (Base‘𝑆)
xpsval.1	⊢ (𝜑 → 𝑅 ∈ 𝑉)
xpsval.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
xpsval.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
xpsval.k	⊢ 𝐺 = (Scalar‘𝑅)
xpsval.u	⊢ 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})

Assertion

Ref	Expression
xpsval	⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))

Distinct variable groups:   𝑥,𝑦   𝑥,𝑊   𝑥,𝑋,𝑦   𝑥,𝑅   𝑥,𝑌,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑦)   𝑆(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑦)

Proof of Theorem xpsval

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsval.t	. 2 ⊢ 𝑇 = (𝑅 ×s 𝑆)
2		xpsval.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
3	2	elexd 2826	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
4		xpsval.2	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
5	4	elexd 2826	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
6		xpsval.x	. . . . . . 7 ⊢ 𝑋 = (Base‘𝑅)
7		basfn 13263	. . . . . . . 8 ⊢ Base Fn V
8		funfvex 5686	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
9	8	funfni 5457	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
10	7, 3, 9	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ∈ V)
11	6, 10	eqeltrid 2319	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
12		xpsval.y	. . . . . . 7 ⊢ 𝑌 = (Base‘𝑆)
13		funfvex 5686	. . . . . . . . 9 ⊢ ((Fun Base ∧ 𝑆 ∈ dom Base) → (Base‘𝑆) ∈ V)
14	13	funfni 5457	. . . . . . . 8 ⊢ ((Base Fn V ∧ 𝑆 ∈ V) → (Base‘𝑆) ∈ V)
15	7, 5, 14	sylancr 414	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑆) ∈ V)
16	12, 15	eqeltrid 2319	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ V)
17		xpsval.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
18	17	mpoexg 6406	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐹 ∈ V)
19	11, 16, 18	syl2anc 411	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
20		cnvexg 5299	. . . . 5 ⊢ (𝐹 ∈ V → ◡𝐹 ∈ V)
21	19, 20	syl 14	. . . 4 ⊢ (𝜑 → ◡𝐹 ∈ V)
22		xpsval.u	. . . . 5 ⊢ 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
23		xpsval.k	. . . . . . 7 ⊢ 𝐺 = (Scalar‘𝑅)
24		scaslid 13358	. . . . . . . . 9 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
25	24	slotex 13231	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (Scalar‘𝑅) ∈ V)
26	2, 25	syl 14	. . . . . . 7 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V)
27	23, 26	eqeltrid 2319	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ V)
28		0lt2o 6673	. . . . . . . 8 ⊢ ∅ ∈ 2o
29		opexg 4343	. . . . . . . 8 ⊢ ((∅ ∈ 2o ∧ 𝑅 ∈ 𝑉) → ⟨∅, 𝑅⟩ ∈ V)
30	28, 2, 29	sylancr 414	. . . . . . 7 ⊢ (𝜑 → ⟨∅, 𝑅⟩ ∈ V)
31		1lt2o 6674	. . . . . . . 8 ⊢ 1o ∈ 2o
32		opexg 4343	. . . . . . . 8 ⊢ ((1o ∈ 2o ∧ 𝑆 ∈ 𝑊) → ⟨1o, 𝑆⟩ ∈ V)
33	31, 4, 32	sylancr 414	. . . . . . 7 ⊢ (𝜑 → ⟨1o, 𝑆⟩ ∈ V)
34		prexg 4324	. . . . . . 7 ⊢ ((⟨∅, 𝑅⟩ ∈ V ∧ ⟨1o, 𝑆⟩ ∈ V) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} ∈ V)
35	30, 33, 34	syl2anc 411	. . . . . 6 ⊢ (𝜑 → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} ∈ V)
36		prdsex 13474	. . . . . 6 ⊢ ((𝐺 ∈ V ∧ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} ∈ V) → (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) ∈ V)
37	27, 35, 36	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) ∈ V)
38	22, 37	eqeltrid 2319	. . . 4 ⊢ (𝜑 → 𝑈 ∈ V)
39		imasex 13510	. . . 4 ⊢ ((◡𝐹 ∈ V ∧ 𝑈 ∈ V) → (◡𝐹 “s 𝑈) ∈ V)
40	21, 38, 39	syl2anc 411	. . 3 ⊢ (𝜑 → (◡𝐹 “s 𝑈) ∈ V)
41		fveq2 5669	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
42	41, 6	eqtr4di 2283	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
43		fveq2 5669	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
44	43, 12	eqtr4di 2283	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
45		mpoeq12 6112	. . . . . . . 8 ⊢ (((Base‘𝑟) = 𝑋 ∧ (Base‘𝑠) = 𝑌) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
46	42, 44, 45	syl2an 289	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
47	46, 17	eqtr4di 2283	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = 𝐹)
48	47	cnveqd 4930	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = ◡𝐹)
49		fveq2 5669	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
50	49	adantr 276	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
51	50, 23	eqtr4di 2283	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
52		simpl 109	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
53	52	opeq2d 3889	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⟨∅, 𝑟⟩ = ⟨∅, 𝑅⟩)
54		simpr 110	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
55	54	opeq2d 3889	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⟨1o, 𝑠⟩ = ⟨1o, 𝑆⟩)
56	53, 55	preq12d 3775	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩} = {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
57	51, 56	oveq12d 6067	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩}) = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
58	57, 22	eqtr4di 2283	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩}) = 𝑈)
59	48, 58	oveq12d 6067	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})) = (◡𝐹 “s 𝑈))
60		df-xps 13509	. . . 4 ⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
61	59, 60	ovmpoga 6182	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ (◡𝐹 “s 𝑈) ∈ V) → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
62	3, 5, 40, 61	syl3anc 1274	. 2 ⊢ (𝜑 → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
63	1, 62	eqtrid 2277	1 ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  Vcvv 2812  ∅c0 3507  {cpr 3689  ⟨cop 3691  ^◡ccnv 4747   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049   ∈ cmpo 6051  1_oc1o 6639  2_oc2o 6640  Basecbs 13204  Scalarcsca 13285  X_scprds 13470   “_s cimas 13504   ×_s cxps 13506

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-1o 6646  df-2o 6647  df-map 6883  df-ixp 6933  df-sup 7274  df-sub 8445  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-dec 9709  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-mulr 13296  df-sca 13298  df-vsca 13299  df-ip 13300  df-tset 13301  df-ple 13302  df-ds 13304  df-hom 13306  df-cco 13307  df-rest 13446  df-topn 13447  df-topgen 13465  df-pt 13466  df-prds 13472  df-iimas 13507  df-xps 13509

This theorem is referenced by: (None)