Theorem xpsle 16241

Description: Value of the ordering in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)

Hypotheses

Ref	Expression
xpsle.t	⊢ 𝑇 = (𝑅 ×s 𝑆)
xpsle.x	⊢ 𝑋 = (Base‘𝑅)
xpsle.y	⊢ 𝑌 = (Base‘𝑆)
xpsle.1	⊢ (𝜑 → 𝑅 ∈ 𝑉)
xpsle.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
xpsle.p	⊢ ≤ = (le‘𝑇)
xpsle.m	⊢ 𝑀 = (le‘𝑅)
xpsle.n	⊢ 𝑁 = (le‘𝑆)
xpsle.3	⊢ (𝜑 → 𝐴 ∈ 𝑋)
xpsle.4	⊢ (𝜑 → 𝐵 ∈ 𝑌)
xpsle.5	⊢ (𝜑 → 𝐶 ∈ 𝑋)
xpsle.6	⊢ (𝜑 → 𝐷 ∈ 𝑌)

Assertion

Ref	Expression
xpsle	⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ≤ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷)))

Proof of Theorem xpsle

Dummy variables 𝑐 𝑑 𝑘 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6653	. . . . 5 ⊢ (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩)
2		xpsle.3	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
3		xpsle.4	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑌)
4		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
5	4	xpsfval 16227	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ◡({𝐴} +𝑐 {𝐵}))
6	2, 3, 5	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐵) = ◡({𝐴} +𝑐 {𝐵}))
7	1, 6	syl5eqr 2670	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}))
8		opelxpi 5148	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
9	2, 3, 8	syl2anc 693	. . . . 5 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
10	4	xpsff1o2 16231	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
11		f1of 6137	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
12	10, 11	ax-mp 5	. . . . . 6 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)⟶ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
13	12	ffvelrni 6358	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
14	9, 13	syl 17	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
15	7, 14	eqeltrrd 2702	. . 3 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
16		df-ov 6653	. . . . 5 ⊢ (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩)
17		xpsle.5	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
18		xpsle.6	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
19	4	xpsfval 16227	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ◡({𝐶} +𝑐 {𝐷}))
20	17, 18, 19	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝐶(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))𝐷) = ◡({𝐶} +𝑐 {𝐷}))
21	16, 20	syl5eqr 2670	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}))
22		opelxpi 5148	. . . . . 6 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
23	17, 18, 22	syl2anc 693	. . . . 5 ⊢ (𝜑 → ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌))
24	12	ffvelrni 6358	. . . . 5 ⊢ (⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
25	23, 24	syl 17	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
26	21, 25	eqeltrrd 2702	. . 3 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
27		xpsle.t	. . . . 5 ⊢ 𝑇 = (𝑅 ×s 𝑆)
28		xpsle.x	. . . . 5 ⊢ 𝑋 = (Base‘𝑅)
29		xpsle.y	. . . . 5 ⊢ 𝑌 = (Base‘𝑆)
30		xpsle.1	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
31		xpsle.2	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
32		eqid 2622	. . . . 5 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
33		eqid 2622	. . . . 5 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
34	27, 28, 29, 30, 31, 4, 32, 33	xpsval 16232	. . . 4 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
35	27, 28, 29, 30, 31, 4, 32, 33	xpslem 16233	. . . 4 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
36		f1ocnv 6149	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
37	10, 36	mp1i 13	. . . . 5 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌))
38		f1ofo 6144	. . . . 5 ⊢ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–1-1-onto→(𝑋 × 𝑌) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌))
39	37, 38	syl 17	. . . 4 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))–onto→(𝑋 × 𝑌))
40		ovexd 6680	. . . 4 ⊢ (𝜑 → ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V)
41		xpsle.p	. . . 4 ⊢ ≤ = (le‘𝑇)
42		eqid 2622	. . . 4 ⊢ (le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
43	37	f1olecpbl 16187	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ 𝑏 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))) ∧ (𝑐 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ 𝑑 ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))) → (((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘𝑎) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘𝑐) ∧ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘𝑏) = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘𝑑)) → (𝑎(le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))𝑏 ↔ 𝑐(le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))𝑑)))
44	34, 35, 39, 40, 41, 42, 43	imasleval 16201	. . 3 ⊢ ((𝜑 ∧ ◡({𝐴} +𝑐 {𝐵}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ ◡({𝐶} +𝑐 {𝐷}) ∈ ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))) → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) ≤ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) ↔ ◡({𝐴} +𝑐 {𝐵})(le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷})))
45	15, 26, 44	mpd3an23 1426	. 2 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) ≤ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) ↔ ◡({𝐴} +𝑐 {𝐵})(le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷})))
46		f1ocnvfv 6534	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩))
47	10, 9, 46	sylancr 695	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐴, 𝐵⟩) = ◡({𝐴} +𝑐 {𝐵}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩))
48	7, 47	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) = ⟨𝐴, 𝐵⟩)
49		f1ocnvfv 6534	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝑋 × 𝑌)) → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩))
50	10, 23, 49	sylancr 695	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘⟨𝐶, 𝐷⟩) = ◡({𝐶} +𝑐 {𝐷}) → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩))
51	21, 50	mpd 15	. . 3 ⊢ (𝜑 → (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) = ⟨𝐶, 𝐷⟩)
52	48, 51	breq12d 4666	. 2 ⊢ (𝜑 → ((◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐴} +𝑐 {𝐵})) ≤ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))‘◡({𝐶} +𝑐 {𝐷})) ↔ ⟨𝐴, 𝐵⟩ ≤ ⟨𝐶, 𝐷⟩))
53		eqid 2622	. . . 4 ⊢ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
54		fvexd 6203	. . . 4 ⊢ (𝜑 → (Scalar‘𝑅) ∈ V)
55		2on 7568	. . . . 5 ⊢ 2𝑜 ∈ On
56	55	a1i 11	. . . 4 ⊢ (𝜑 → 2𝑜 ∈ On)
57		xpscfn 16219	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
58	30, 31, 57	syl2anc 693	. . . 4 ⊢ (𝜑 → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
59	15, 35	eleqtrd 2703	. . . 4 ⊢ (𝜑 → ◡({𝐴} +𝑐 {𝐵}) ∈ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
60	26, 35	eleqtrd 2703	. . . 4 ⊢ (𝜑 → ◡({𝐶} +𝑐 {𝐷}) ∈ (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
61	33, 53, 54, 56, 58, 59, 60, 42	prdsleval 16137	. . 3 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷}) ↔ ∀𝑘 ∈ 2𝑜 (◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘)))
62		df2o3 7573	. . . . . 6 ⊢ 2𝑜 = {∅, 1𝑜}
63	62	raleqi 3142	. . . . 5 ⊢ (∀𝑘 ∈ 2𝑜 (◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘) ↔ ∀𝑘 ∈ {∅, 1𝑜} (◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘))
64		0ex 4790	. . . . . 6 ⊢ ∅ ∈ V
65		1on 7567	. . . . . . 7 ⊢ 1𝑜 ∈ On
66	65	elexi 3213	. . . . . 6 ⊢ 1𝑜 ∈ V
67		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = ∅ → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = (◡({𝐴} +𝑐 {𝐵})‘∅))
68		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = ∅ → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = (◡({𝑅} +𝑐 {𝑆})‘∅))
69	68	fveq2d 6195	. . . . . . 7 ⊢ (𝑘 = ∅ → (le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (le‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
70		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = ∅ → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = (◡({𝐶} +𝑐 {𝐷})‘∅))
71	67, 69, 70	breq123d 4667	. . . . . 6 ⊢ (𝑘 = ∅ → ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘) ↔ (◡({𝐴} +𝑐 {𝐵})‘∅)(le‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅)))
72		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = 1𝑜 → (◡({𝐴} +𝑐 {𝐵})‘𝑘) = (◡({𝐴} +𝑐 {𝐵})‘1𝑜))
73		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = 1𝑜 → (◡({𝑅} +𝑐 {𝑆})‘𝑘) = (◡({𝑅} +𝑐 {𝑆})‘1𝑜))
74	73	fveq2d 6195	. . . . . . 7 ⊢ (𝑘 = 1𝑜 → (le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘)) = (le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
75		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = 1𝑜 → (◡({𝐶} +𝑐 {𝐷})‘𝑘) = (◡({𝐶} +𝑐 {𝐷})‘1𝑜))
76	72, 74, 75	breq123d 4667	. . . . . 6 ⊢ (𝑘 = 1𝑜 → ((◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘) ↔ (◡({𝐴} +𝑐 {𝐵})‘1𝑜)(le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)))
77	64, 66, 71, 76	ralpr 4238	. . . . 5 ⊢ (∀𝑘 ∈ {∅, 1𝑜} (◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘) ↔ ((◡({𝐴} +𝑐 {𝐵})‘∅)(le‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅) ∧ (◡({𝐴} +𝑐 {𝐵})‘1𝑜)(le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)))
78	63, 77	bitri 264	. . . 4 ⊢ (∀𝑘 ∈ 2𝑜 (◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘) ↔ ((◡({𝐴} +𝑐 {𝐵})‘∅)(le‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅) ∧ (◡({𝐴} +𝑐 {𝐵})‘1𝑜)(le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)))
79		xpsc0 16220	. . . . . . 7 ⊢ (𝐴 ∈ 𝑋 → (◡({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)
80	2, 79	syl 17	. . . . . 6 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})‘∅) = 𝐴)
81		xpsc0 16220	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
82	30, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
83	82	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (le‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = (le‘𝑅))
84		xpsle.m	. . . . . . 7 ⊢ 𝑀 = (le‘𝑅)
85	83, 84	syl6eqr 2674	. . . . . 6 ⊢ (𝜑 → (le‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = 𝑀)
86		xpsc0 16220	. . . . . . 7 ⊢ (𝐶 ∈ 𝑋 → (◡({𝐶} +𝑐 {𝐷})‘∅) = 𝐶)
87	17, 86	syl 17	. . . . . 6 ⊢ (𝜑 → (◡({𝐶} +𝑐 {𝐷})‘∅) = 𝐶)
88	80, 85, 87	breq123d 4667	. . . . 5 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘∅)(le‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅) ↔ 𝐴𝑀𝐶))
89		xpsc1 16221	. . . . . . 7 ⊢ (𝐵 ∈ 𝑌 → (◡({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)
90	3, 89	syl 17	. . . . . 6 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})‘1𝑜) = 𝐵)
91		xpsc1 16221	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝑊 → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
92	31, 91	syl 17	. . . . . . . 8 ⊢ (𝜑 → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
93	92	fveq2d 6195	. . . . . . 7 ⊢ (𝜑 → (le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = (le‘𝑆))
94		xpsle.n	. . . . . . 7 ⊢ 𝑁 = (le‘𝑆)
95	93, 94	syl6eqr 2674	. . . . . 6 ⊢ (𝜑 → (le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = 𝑁)
96		xpsc1 16221	. . . . . . 7 ⊢ (𝐷 ∈ 𝑌 → (◡({𝐶} +𝑐 {𝐷})‘1𝑜) = 𝐷)
97	18, 96	syl 17	. . . . . 6 ⊢ (𝜑 → (◡({𝐶} +𝑐 {𝐷})‘1𝑜) = 𝐷)
98	90, 95, 97	breq123d 4667	. . . . 5 ⊢ (𝜑 → ((◡({𝐴} +𝑐 {𝐵})‘1𝑜)(le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜) ↔ 𝐵𝑁𝐷))
99	88, 98	anbi12d 747	. . . 4 ⊢ (𝜑 → (((◡({𝐴} +𝑐 {𝐵})‘∅)(le‘(◡({𝑅} +𝑐 {𝑆})‘∅))(◡({𝐶} +𝑐 {𝐷})‘∅) ∧ (◡({𝐴} +𝑐 {𝐵})‘1𝑜)(le‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜))(◡({𝐶} +𝑐 {𝐷})‘1𝑜)) ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷)))
100	78, 99	syl5bb 272	. . 3 ⊢ (𝜑 → (∀𝑘 ∈ 2𝑜 (◡({𝐴} +𝑐 {𝐵})‘𝑘)(le‘(◡({𝑅} +𝑐 {𝑆})‘𝑘))(◡({𝐶} +𝑐 {𝐷})‘𝑘) ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷)))
101	61, 100	bitrd 268	. 2 ⊢ (𝜑 → (◡({𝐴} +𝑐 {𝐵})(le‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))◡({𝐶} +𝑐 {𝐷}) ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷)))
102	45, 52, 101	3bitr3d 298	1 ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ≤ ⟨𝐶, 𝐷⟩ ↔ (𝐴𝑀𝐶 ∧ 𝐵𝑁𝐷)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ^◡ccnv 5113  ran crn 5115  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑐 ccda 8989  Basecbs 15857  Scalarcsca 15944  lecple 15948  X_scprds 16106   ×_s cxps 16166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108  df-imas 16168  df-xps 16170

This theorem is referenced by: (None)