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Theorem gtiso 32612
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
StepHypRef Expression
1 eqid 2726 . . . . 5 ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2 eqid 2726 . . . . 5 ((𝐵 × 𝐵) ∖ < ) = ((𝐵 × 𝐵) ∖ < )
31, 2isocnv3 7344 . . . 4 (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵))
43a1i 11 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
5 df-le 11304 . . . . . . . . . 10 ≤ = ((ℝ* × ℝ*) ∖ < )
65cnveqi 5881 . . . . . . . . 9 ≤ = ((ℝ* × ℝ*) ∖ < )
7 cnvdif 6155 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
8 cnvxp 6168 . . . . . . . . . 10 (ℝ* × ℝ*) = (ℝ* × ℝ*)
9 ltrel 11326 . . . . . . . . . . 11 Rel <
10 dfrel2 6200 . . . . . . . . . . 11 (Rel < ↔ < = < )
119, 10mpbi 229 . . . . . . . . . 10 < = <
128, 11difeq12i 4119 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
136, 7, 123eqtri 2758 . . . . . . . 8 ≤ = ((ℝ* × ℝ*) ∖ < )
1413ineq1i 4209 . . . . . . 7 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴))
15 indif1 4273 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
1614, 15eqtri 2754 . . . . . 6 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
17 xpss12 5697 . . . . . . . . 9 ((𝐴 ⊆ ℝ*𝐴 ⊆ ℝ*) → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
1817anidms 565 . . . . . . . 8 (𝐴 ⊆ ℝ* → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
19 sseqin2 4216 . . . . . . . 8 ((𝐴 × 𝐴) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2018, 19sylib 217 . . . . . . 7 (𝐴 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2120difeq1d 4120 . . . . . 6 (𝐴 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
2216, 21eqtr2id 2779 . . . . 5 (𝐴 ⊆ ℝ* → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
2322adantr 479 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
24 isoeq2 7330 . . . 4 (((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
2523, 24syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
265ineq1i 4209 . . . . . . 7 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵))
27 indif1 4273 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
2826, 27eqtri 2754 . . . . . 6 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
29 xpss12 5697 . . . . . . . . 9 ((𝐵 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
3029anidms 565 . . . . . . . 8 (𝐵 ⊆ ℝ* → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
31 sseqin2 4216 . . . . . . . 8 ((𝐵 × 𝐵) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3230, 31sylib 217 . . . . . . 7 (𝐵 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3332difeq1d 4120 . . . . . 6 (𝐵 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < ) = ((𝐵 × 𝐵) ∖ < ))
3428, 33eqtr2id 2779 . . . . 5 (𝐵 ⊆ ℝ* → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
3534adantl 480 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36 isoeq3 7331 . . . 4 (((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
3735, 36syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
384, 25, 373bitrd 304 . 2 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39 isocnv2 7343 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
40 isores2 7345 . . . 4 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41 isores1 7346 . . . 4 (𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4240, 41bitri 274 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43 lerel 11328 . . . . 5 Rel ≤
44 dfrel2 6200 . . . . 5 (Rel ≤ ↔ ≤ = ≤ )
4543, 44mpbi 229 . . . 4 ≤ = ≤
46 isoeq2 7330 . . . 4 ( ≤ = ≤ → (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
4745, 46ax-mp 5 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
4839, 42, 473bitr3ri 301 . 2 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4938, 48bitr4di 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 394   = wceq 1534  cdif 3944  cin 3946  wss 3947   × cxp 5680  ccnv 5681  Rel wrel 5687   Isom wiso 6555  *cxr 11297   < clt 11298  cle 11299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-xr 11302  df-ltxr 11303  df-le 11304
This theorem is referenced by:  xrge0iifhmeo  33751
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