gtiso - Mathbox for Thierry Arnoux

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Theorem gtiso 31614

Description: Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)

**Assertion**
Ref	Expression
gtiso	⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))

**Proof of Theorem gtiso**
Step	Hyp	Ref	Expression
1		eqid 2736	. . . . 5 ⊢ ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2		eqid 2736	. . . . 5 ⊢ ((𝐵 × 𝐵) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < )
3	1, 2	isocnv3 7277	. . . 4 ⊢ (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵))
4	3	a1i 11	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
5		df-le 11195	. . . . . . . . . 10 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
6	5	cnveqi 5830	. . . . . . . . 9 ⊢ ^◡ ≤ = ^◡((ℝ^* × ℝ^*) ∖ ^◡ < )
7		cnvdif 6096	. . . . . . . . 9 ⊢ ^◡((ℝ^* × ℝ^) ∖ ^◡ < ) = (^◡(ℝ^ × ℝ^*) ∖ ^◡^◡ < )
8		cnvxp 6109	. . . . . . . . . 10 ⊢ ^◡(ℝ^* × ℝ^) = (ℝ^ × ℝ^*)
9		ltrel 11217	. . . . . . . . . . 11 ⊢ Rel <
10		dfrel2 6141	. . . . . . . . . . 11 ⊢ (Rel < ↔ ^◡^◡ < = < )
11	9, 10	mpbi 229	. . . . . . . . . 10 ⊢ ^◡^◡ < = <
12	8, 11	difeq12i 4080	. . . . . . . . 9 ⊢ (^◡(ℝ^* × ℝ^) ∖ ^◡^◡ < ) = ((ℝ^ × ℝ^*) ∖ < )
13	6, 7, 12	3eqtri 2768	. . . . . . . 8 ⊢ ^◡ ≤ = ((ℝ^* × ℝ^*) ∖ < )
14	13	ineq1i 4168	. . . . . . 7 ⊢ (^◡ ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∖ < ) ∩ (𝐴 × 𝐴))
15		indif1 4231	. . . . . . 7 ⊢ (((ℝ^* × ℝ^) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < )
16	14, 15	eqtri 2764	. . . . . 6 ⊢ (^◡ ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < )
17		xpss12 5648	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^) → (𝐴 × 𝐴) ⊆ (ℝ^ × ℝ^*))
18	17	anidms 567	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^*))
19		sseqin2 4175	. . . . . . . 8 ⊢ ((𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
20	18, 19	sylib 217	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
21	20	difeq1d 4081	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
22	16, 21	eqtr2id 2789	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → ((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)))
23	22	adantr 481	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → ((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)))
24		isoeq2 7263	. . . 4 ⊢ (((𝐴 × 𝐴) ∖ < ) = (^◡ ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
25	23, 24	syl 17	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
26	5	ineq1i 4168	. . . . . . 7 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∖ ^◡ < ) ∩ (𝐵 × 𝐵))
27		indif1 4231	. . . . . . 7 ⊢ (((ℝ^* × ℝ^) ∖ ^◡ < ) ∩ (𝐵 × 𝐵)) = (((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
28	26, 27	eqtri 2764	. . . . . 6 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
29		xpss12 5648	. . . . . . . . 9 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐵 × 𝐵) ⊆ (ℝ^ × ℝ^*))
30	29	anidms 567	. . . . . . . 8 ⊢ (𝐵 ⊆ ℝ^* → (𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^*))
31		sseqin2 4175	. . . . . . . 8 ⊢ ((𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
32	30, 31	sylib 217	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
33	32	difeq1d 4081	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < ))
34	28, 33	eqtr2id 2789	. . . . 5 ⊢ (𝐵 ⊆ ℝ^* → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
35	34	adantl 482	. . . 4 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36		isoeq3 7264	. . . 4 ⊢ (((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
37	35, 36	syl 17	. . 3 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
38	4, 25, 37	3bitrd 304	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39		isocnv2 7276	. . 3 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵))
40		isores2 7278	. . . 4 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ^◡ ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41		isores1 7279	. . . 4 ⊢ (𝐹 Isom ^◡ ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
42	40, 41	bitri 274	. . 3 ⊢ (𝐹 Isom ^◡ ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43		lerel 11219	. . . . 5 ⊢ Rel ≤
44		dfrel2 6141	. . . . 5 ⊢ (Rel ≤ ↔ ^◡^◡ ≤ = ≤ )
45	43, 44	mpbi 229	. . . 4 ⊢ ^◡^◡ ≤ = ≤
46		isoeq2 7263	. . . 4 ⊢ (^◡^◡ ≤ = ≤ → (𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))
47	45, 46	ax-mp 5	. . 3 ⊢ (𝐹 Isom ^◡^◡ ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵))
48	39, 42, 47	3bitr3ri 301	. 2 ⊢ (𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵) ↔ 𝐹 Isom (^◡ ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
49	38, 48	bitr4di 288	1 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ^◡ ≤ (𝐴, 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∖ cdif 3907 ∩ cin 3909 ⊆ wss 3910 × cxp 5631 ^◡ccnv 5632 Rel wrel 5638 Isom wiso 6497 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190

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

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-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 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-br 5106 df-opab 5168 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-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-xr 11193 df-ltxr 11194 df-le 11195

This theorem is referenced by: xrge0iifhmeo 32517

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