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Mirrors > Home > MPE Home > Th. List > 2sqreulem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for 2sqreu 27407 et. (Contributed by AV, 25-Jun-2023.) |
Ref | Expression |
---|---|
2sqreulem4.1 | ⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
Ref | Expression |
---|---|
2sqreulem4 | ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sqreulem3 27404 | . . . 4 ⊢ ((𝑎 ∈ ℕ0 ∧ (𝑏 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0)) → (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) | |
2 | 1 | ralrimivva 3191 | . . 3 ⊢ (𝑎 ∈ ℕ0 → ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
3 | 2sqreulem4.1 | . . . . 5 ⊢ (𝜑 ↔ (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) | |
4 | 3 | rmobii 3372 | . . . 4 ⊢ (∃*𝑏 ∈ ℕ0 𝜑 ↔ ∃*𝑏 ∈ ℕ0 (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃)) |
5 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑏ℕ0 | |
6 | nfcv 2892 | . . . . 5 ⊢ Ⅎ𝑐ℕ0 | |
7 | nfsbc1v 3788 | . . . . . 6 ⊢ Ⅎ𝑏[𝑐 / 𝑏]𝜓 | |
8 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑏((𝑎↑2) + (𝑐↑2)) = 𝑃 | |
9 | 7, 8 | nfan 1894 | . . . . 5 ⊢ Ⅎ𝑏([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃) |
10 | sbceq1a 3779 | . . . . . 6 ⊢ (𝑏 = 𝑐 → (𝜓 ↔ [𝑐 / 𝑏]𝜓)) | |
11 | oveq1 7423 | . . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝑏↑2) = (𝑐↑2)) | |
12 | 11 | oveq2d 7432 | . . . . . . 7 ⊢ (𝑏 = 𝑐 → ((𝑎↑2) + (𝑏↑2)) = ((𝑎↑2) + (𝑐↑2))) |
13 | 12 | eqeq1d 2727 | . . . . . 6 ⊢ (𝑏 = 𝑐 → (((𝑎↑2) + (𝑏↑2)) = 𝑃 ↔ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) |
14 | 10, 13 | anbi12d 630 | . . . . 5 ⊢ (𝑏 = 𝑐 → ((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃))) |
15 | 5, 6, 9, 14 | rmo4f 3722 | . . . 4 ⊢ (∃*𝑏 ∈ ℕ0 (𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ↔ ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
16 | 4, 15 | bitri 274 | . . 3 ⊢ (∃*𝑏 ∈ ℕ0 𝜑 ↔ ∀𝑏 ∈ ℕ0 ∀𝑐 ∈ ℕ0 (((𝜓 ∧ ((𝑎↑2) + (𝑏↑2)) = 𝑃) ∧ ([𝑐 / 𝑏]𝜓 ∧ ((𝑎↑2) + (𝑐↑2)) = 𝑃)) → 𝑏 = 𝑐)) |
17 | 2, 16 | sylibr 233 | . 2 ⊢ (𝑎 ∈ ℕ0 → ∃*𝑏 ∈ ℕ0 𝜑) |
18 | 17 | rgen 3053 | 1 ⊢ ∀𝑎 ∈ ℕ0 ∃*𝑏 ∈ ℕ0 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃*wrmo 3363 [wsbc 3768 (class class class)co 7416 + caddc 11141 2c2 12297 ℕ0cn0 12502 ↑cexp 14058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-seq 13999 df-exp 14059 |
This theorem is referenced by: 2sqreunnlem2 27406 2sqreu 27407 2sqreult 27409 2sqreultb 27410 |
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