Theorem ismtycnv 35082

Description: The inverse of an isometry is an isometry. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
ismtycnv	⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀)))

Proof of Theorem ismtycnv

Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1ocnv 6629	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
2	1	adantr 483	. . . 4 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ◡𝐹:𝑌–1-1-onto→𝑋)
3		f1ocnvdm 7043	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (◡𝐹‘𝑢) ∈ 𝑋)
4	3	ex 415	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑢 ∈ 𝑌 → (◡𝐹‘𝑢) ∈ 𝑋))
5		f1ocnvdm 7043	. . . . . . . . . . 11 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (◡𝐹‘𝑣) ∈ 𝑋)
6	5	ex 415	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → (𝑣 ∈ 𝑌 → (◡𝐹‘𝑣) ∈ 𝑋))
7	4, 6	anim12d 610	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)))
8	7	adantr 483	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ((𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌) → ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)))
9	8	imdistani 571	. . . . . . 7 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)))
10		oveq1 7165	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝑥𝑀𝑦) = ((◡𝐹‘𝑢)𝑀𝑦))
11		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑥 = (◡𝐹‘𝑢) → (𝐹‘𝑥) = (𝐹‘(◡𝐹‘𝑢)))
12	11	oveq1d 7173	. . . . . . . . . . 11 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)))
13	10, 12	eqeq12d 2839	. . . . . . . . . 10 ⊢ (𝑥 = (◡𝐹‘𝑢) → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦))))
14		oveq2 7166	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((◡𝐹‘𝑢)𝑀𝑦) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
15		fveq2 6672	. . . . . . . . . . . 12 ⊢ (𝑦 = (◡𝐹‘𝑣) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑣)))
16	15	oveq2d 7174	. . . . . . . . . . 11 ⊢ (𝑦 = (◡𝐹‘𝑣) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
17	14, 16	eqeq12d 2839	. . . . . . . . . 10 ⊢ (𝑦 = (◡𝐹‘𝑣) → (((◡𝐹‘𝑢)𝑀𝑦) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘𝑦)) ↔ ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))))
18	13, 17	rspc2v 3635	. . . . . . . . 9 ⊢ (((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣)))))
19	18	impcom 410	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
20	19	adantll 712	. . . . . . 7 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ ((◡𝐹‘𝑢) ∈ 𝑋 ∧ (◡𝐹‘𝑣) ∈ 𝑋)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
21	9, 20	syl 17	. . . . . 6 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)) = ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))))
22		f1ocnvfv2 7036	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑢 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
23	22	adantrr 715	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑢)) = 𝑢)
24		f1ocnvfv2 7036	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑣 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
25	24	adantrl 714	. . . . . . . 8 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝐹‘(◡𝐹‘𝑣)) = 𝑣)
26	23, 25	oveq12d 7176	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣))
27	26	adantlr 713	. . . . . 6 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑢))𝑁(𝐹‘(◡𝐹‘𝑣))) = (𝑢𝑁𝑣))
28	21, 27	eqtr2d 2859	. . . . 5 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
29	28	ralrimivva 3193	. . . 4 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))
30	2, 29	jca 514	. . 3 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣))))
31	30	a1i 11	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))))
32		isismty 35081	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))))
33		isismty 35081	. . 3 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑀 ∈ (∞Met‘𝑋)) → (◡𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))))
34	33	ancoms 461	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (◡𝐹 ∈ (𝑁 Ismty 𝑀) ↔ (◡𝐹:𝑌–1-1-onto→𝑋 ∧ ∀𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 (𝑢𝑁𝑣) = ((◡𝐹‘𝑢)𝑀(◡𝐹‘𝑣)))))
35	31, 32, 34	3imtr4d 296	1 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ^◡ccnv 5556  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  ∞Metcxmet 20532   Ismty cismty 35078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-map 8410  df-xr 10681  df-xmet 20540  df-ismty 35079

This theorem is referenced by:  ismtyhmeolem  35084  ismtyhmeo  35085  ismtybnd  35087