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Theorem prdsdsval 17441

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

**Hypotheses**
Ref	Expression
prdsbasmpt.y	⊢ 𝑌 = (𝑆X_s𝑅)
prdsbasmpt.b	⊢ 𝐵 = (Base‘𝑌)
prdsbasmpt.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbasmpt.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdsbasmpt.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdsplusgval.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
prdsplusgval.g	⊢ (𝜑 → 𝐺 ∈ 𝐵)
prdsdsval.d	⊢ 𝐷 = (dist‘𝑌)

**Assertion**
Ref	Expression
prdsdsval	⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐹 𝑥,𝐺 𝜑,𝑥 𝑥,𝐼 𝑥,𝑉 𝑥,𝑅 𝑥,𝑆 𝑥,𝑊 𝑥,𝑌 Allowed substitution hint: 𝐷(𝑥)

Proof of Theorem prdsdsval Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbasmpt.y	. . 3 ⊢ 𝑌 = (𝑆X_s𝑅)
2		prdsbasmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		prdsbasmpt.r	. . . 4 ⊢ (𝜑 → 𝑅 Fn 𝐼)
4		prdsbasmpt.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
5		fnex 7172	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 585	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
7		prdsbasmpt.b	. . 3 ⊢ 𝐵 = (Base‘𝑌)
8		fndm 6601	. . . 4 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
9	3, 8	syl 17	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
10		prdsdsval.d	. . 3 ⊢ 𝐷 = (dist‘𝑌)
11	1, 2, 6, 7, 9, 10	prdsds 17427	. 2 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < )))
12		fveq1 6839	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 6839	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
14	12, 13	oveqan12d 7386	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))
15	14	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))
16	15	mpteq2dv 5179	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))))
17	16	rneqd 5893	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))))
18	17	uneq1d 4107	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}) = (ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}))
19	18	supeq1d 9359	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^, < ) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^, < ))
20		prdsplusgval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
21		prdsplusgval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
22		xrltso 13092	. . . 4 ⊢ < Or ℝ^*
23	22	supex 9377	. . 3 ⊢ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ) ∈ V
24	23	a1i 11	. 2 ⊢ (𝜑 → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ) ∈ V)
25	11, 19, 20, 21, 24	ovmpod 7519	1 ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3429 ∪ cun 3887 {csn 4567 ↦ cmpt 5166 dom cdm 5631 ran crn 5632 Fn wfn 6493 ‘cfv 6498 (class class class)co 7367 supcsup 9353 0cc0 11038 ℝ^*cxr 11178 < clt 11179 Basecbs 17179 distcds 17229 X_scprds 17408

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-hom 17244 df-cco 17245 df-prds 17410

This theorem is referenced by: prdsdsval2 17447 xpsdsval 24346

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