leiso - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > leiso		Structured version Visualization version GIF version

Theorem leiso 14420

Description: Two ways to write a strictly increasing function on the reals. (Contributed by Mario Carneiro, 9-Sep-2015.)

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

**Proof of Theorem leiso**
Step	Hyp	Ref	Expression
1		df-le 11254	. . . . . . 7 ⊢ ≤ = ((ℝ^* × ℝ^*) ∖ ^◡ < )
2	1	ineq1i 4209	. . . . . 6 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∖ ^◡ < ) ∩ (𝐴 × 𝐴))
3		indif1 4272	. . . . . 6 ⊢ (((ℝ^* × ℝ^) ∖ ^◡ < ) ∩ (𝐴 × 𝐴)) = (((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ ^◡ < )
4	2, 3	eqtri 2761	. . . . 5 ⊢ ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ ^◡ < )
5		xpss12 5692	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^) → (𝐴 × 𝐴) ⊆ (ℝ^ × ℝ^*))
6	5	anidms 568	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → (𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^*))
7		sseqin2 4216	. . . . . . 7 ⊢ ((𝐴 × 𝐴) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
8	6, 7	sylib 217	. . . . . 6 ⊢ (𝐴 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9	8	difeq1d 4122	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐴 × 𝐴)) ∖ ^◡ < ) = ((𝐴 × 𝐴) ∖ ^◡ < ))
10	4, 9	eqtr2id 2786	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → ((𝐴 × 𝐴) ∖ ^◡ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
11		isoeq2 7315	. . . 4 ⊢ (((𝐴 × 𝐴) ∖ ^◡ < ) = ( ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
12	10, 11	syl 17	. . 3 ⊢ (𝐴 ⊆ ℝ^* → (𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵)))
13	1	ineq1i 4209	. . . . . 6 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∖ ^◡ < ) ∩ (𝐵 × 𝐵))
14		indif1 4272	. . . . . 6 ⊢ (((ℝ^* × ℝ^) ∖ ^◡ < ) ∩ (𝐵 × 𝐵)) = (((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
15	13, 14	eqtri 2761	. . . . 5 ⊢ ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < )
16		xpss12 5692	. . . . . . . 8 ⊢ ((𝐵 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^) → (𝐵 × 𝐵) ⊆ (ℝ^ × ℝ^*))
17	16	anidms 568	. . . . . . 7 ⊢ (𝐵 ⊆ ℝ^* → (𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^*))
18		sseqin2 4216	. . . . . . 7 ⊢ ((𝐵 × 𝐵) ⊆ (ℝ^* × ℝ^) ↔ ((ℝ^ × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
19	17, 18	sylib 217	. . . . . 6 ⊢ (𝐵 ⊆ ℝ^* → ((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
20	19	difeq1d 4122	. . . . 5 ⊢ (𝐵 ⊆ ℝ^* → (((ℝ^* × ℝ^*) ∩ (𝐵 × 𝐵)) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < ))
21	15, 20	eqtr2id 2786	. . . 4 ⊢ (𝐵 ⊆ ℝ^* → ((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
22		isoeq3 7316	. . . 4 ⊢ (((𝐵 × 𝐵) ∖ ^◡ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
23	21, 22	syl 17	. . 3 ⊢ (𝐵 ⊆ ℝ^* → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
24	12, 23	sylan9bb 511	. 2 ⊢ ((𝐴 ⊆ ℝ^* ∧ 𝐵 ⊆ ℝ^*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
25		isocnv2 7328	. . 3 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ^◡ < , ^◡ < (𝐴, 𝐵))
26		eqid 2733	. . . 4 ⊢ ((𝐴 × 𝐴) ∖ ^◡ < ) = ((𝐴 × 𝐴) ∖ ^◡ < )
27		eqid 2733	. . . 4 ⊢ ((𝐵 × 𝐵) ∖ ^◡ < ) = ((𝐵 × 𝐵) ∖ ^◡ < )
28	26, 27	isocnv3 7329	. . 3 ⊢ (𝐹 Isom ^◡ < , ^◡ < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵))
29	25, 28	bitri 275	. 2 ⊢ (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ ^◡ < ), ((𝐵 × 𝐵) ∖ ^◡ < )(𝐴, 𝐵))
30		isores1 7331	. . 3 ⊢ (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ≤ (𝐴, 𝐵))
31		isores2 7330	. . 3 ⊢ (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
32	30, 31	bitri 275	. 2 ⊢ (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
33	24, 29, 32	3bitr4g 314	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 397 = wceq 1542 ∖ cdif 3946 ∩ cin 3948 ⊆ wss 3949 × cxp 5675 ^◡ccnv 5676 Isom wiso 6545 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-le 11254

This theorem is referenced by: leisorel 14421 icopnfhmeo 24459 iccpnfhmeo 24461 xrhmeo 24462

W3C validator